Theorem adjvalt 9799

Description: Value of the adjoint function. By adjmo 9743 and euuni 2881, the union should be read "the unique u such that..." for T in the domain of adj_h.

Assertion

Ref	Expression
adjvalt	|- (T:H~-->H~ -> (adjh` T) = U.{u | (u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))})

Distinct variable group:   x,u,y,T

Proof of Theorem adjvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8854	. . . 4 |- H~ e. V
2		fex 3652	. . . 4 |- ((T:H~-->H~ /\ H~ e. V) -> T e. V)
3	1, 2	mpan2 696	. . 3 |- (T:H~-->H~ -> T e. V)
4		funadj 9798	. . . 4 |- Fun adjh
5		dfadj2 9797	. . . . 5 |- adjh = {<.t, u>. | (t:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (t` y)) = ((u` x) .ih y))}
6		feq1 3620	. . . . . 6 |- (t = T -> (t:H~-->H~ <-> T:H~-->H~))
7		fveq1 3723	. . . . . . . . 9 |- (t = T -> (t` y) = (T` y))
8	7	opreq2d 3976	. . . . . . . 8 |- (t = T -> (x .ih (t` y)) = (x .ih (T` y)))
9	8	eqeq1d 1483	. . . . . . 7 |- (t = T -> ((x .ih (t` y)) = ((u` x) .ih y) <-> (x .ih (T` y)) = ((u` x) .ih y)))
10	9	2ralbidv 1680	. . . . . 6 |- (t = T -> (A.x e. H~ A.y e. H~ (x .ih (t` y)) = ((u` x) .ih y) <-> A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y)))
11	6, 10	3anbi13d 895	. . . . 5 |- (t = T -> ((t:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (t` y)) = ((u` x) .ih y)) <-> (T:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))))
12	5, 11	fvopab5 3793	. . . 4 |- ((Fun adjh /\ T e. V) -> (adjh` T) = U.{u | (T:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))})
13	4, 12	mpan 695	. . 3 |- (T e. V -> (adjh` T) = U.{u | (T:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))})
14	3, 13	syl 10	. 2 |- (T:H~-->H~ -> (adjh` T) = U.{u | (T:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))})
15		3anass 779	. . . . 5 |- ((T:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y)) <-> (T:H~-->H~ /\ (u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))))
16	15	baib 685	. . . 4 |- (T:H~-->H~ -> ((T:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y)) <-> (u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))))
17	16	abbidv 1577	. . 3 |- (T:H~-->H~ -> {u | (T:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))} = {u | (u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))})
18	17	unieqd 2512	. 2 |- (T:H~-->H~ -> U.{u | (T:H~-->H~ /\ u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))} = U.{u | (u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))})
19	14, 18	eqtrd 1507	1 |- (T:H~-->H~ -> (adjh` T) = U.{u | (u:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((u` x) .ih y))})

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  Vcvv 1811  U.cuni 2503  Fun wfun 3176  -->wf 3178  ` cfv 3182  (class class class)co 3963  H~chil 8773   .ih csp 8778  adj_hcado 8809

This theorem is referenced by:  adjval2t 9800

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4617  ax-hilex 8854  ax-hfvadd 8855  ax-hvcom 8856  ax-hvass 8857  ax-hv0cl 8858  ax-hvaddid 8859  ax-hfvmul 8860  ax-hvmulid 8861  ax-hvdistr2 8864  ax-hvmul0 8865  ax-hfi 8931  ax-his1 8934  ax-his2 8935  ax-his3 8936  ax-his4 8937

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 4992  df-pli 4993  df-mi 4994  df-lti 4995  df-plpq 5027  df-mpq 5028  df-enq 5029  df-nq 5030  df-plq 5031  df-mq 5032  df-rq 5033  df-ltq 5034  df-1q 5035  df-np 5078  df-1p 5079  df-plp 5080  df-mp 5081  df-ltp 5082  df-plpr 5156  df-mpr 5157  df-enr 5158  df-nr 5159  df-plr 5160  df-mr 5161  df-ltr 5162  df-0r 5163  df-1r 5164  df-m1r 5165  df-c 5232  df-0 5233  df-1 5234  df-i 5235  df-r 5236  df-plus 5237  df-mul 5238  df-lt 5239  df-sub 5348  df-neg 5350  df-pnf 5479  df-mnf 5480  df-xr 5481  df-ltxr 5482  df-le 5483  df-div 5691  df-re 6737  df-im 6738  df-cj 6739  df-hvsub 8825  df-adjh 9760

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