Theorem fvsnun2 3812

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 3811.

Hypotheses

Ref	Expression
fvsnun.1	|- A e. V
fvsnun.2	|- B e. V
fvsnun.3	|- G = ({<.A, B>.} u. (F |` (C \ {A})))

Assertion

Ref	Expression
fvsnun2	|- (D e. (C \ {A}) -> (G` D) = (F` D))

Proof of Theorem fvsnun2

Step	Hyp	Ref	Expression
1		fvres 3750	. 2 |- (D e. (C \ {A}) -> ((G |` (C \ {A}))` D) = (G` D))
2		fvres 3750	. . 3 |- (D e. (C \ {A}) -> ((F |` (C \ {A}))` D) = (F` D))
3		fvsnun.3	. . . . . 6 |- G = ({<.A, B>.} u. (F |` (C \ {A})))
4		reseq1 3384	. . . . . 6 |- (G = ({<.A, B>.} u. (F |` (C \ {A}))) -> (G |` (C \ {A})) = (({<.A, B>.} u. (F |` (C \ {A}))) |` (C \ {A})))
5	3, 4	ax-mp 7	. . . . 5 |- (G |` (C \ {A})) = (({<.A, B>.} u. (F |` (C \ {A}))) |` (C \ {A}))
6		resundir 3395	. . . . 5 |- (({<.A, B>.} u. (F |` (C \ {A}))) |` (C \ {A})) = (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A})))
7		difdisj 2349	. . . . . . . 8 |- ({A} i^i (C \ {A})) = (/)
8		fvsnun.1	. . . . . . . . . . 11 |- A e. V
9		fvsnun.2	. . . . . . . . . . 11 |- B e. V
10	8, 9	f1osn 3735	. . . . . . . . . 10 |- {<.A, B>.}:{A}-1-1-onto->{B}
11		f1ofn 3706	. . . . . . . . . 10 |- ({<.A, B>.}:{A}-1-1-onto->{B} -> {<.A, B>.} Fn {A})
12	10, 11	ax-mp 7	. . . . . . . . 9 |- {<.A, B>.} Fn {A}
13		fnresdisj 3613	. . . . . . . . 9 |- ({<.A, B>.} Fn {A} -> (({A} i^i (C \ {A})) = (/) <-> ({<.A, B>.} |` (C \ {A})) = (/)))
14	12, 13	ax-mp 7	. . . . . . . 8 |- (({A} i^i (C \ {A})) = (/) <-> ({<.A, B>.} |` (C \ {A})) = (/))
15	7, 14	mpbi 189	. . . . . . 7 |- ({<.A, B>.} |` (C \ {A})) = (/)
16		residm 3406	. . . . . . 7 |- ((F |` (C \ {A})) |` (C \ {A})) = (F |` (C \ {A}))
17	15, 16	uneq12i 2193	. . . . . 6 |- (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A}))) = ((/) u. (F |` (C \ {A})))
18		uncom 2187	. . . . . 6 |- ((/) u. (F |` (C \ {A}))) = ((F |` (C \ {A})) u. (/))
19		un0 2309	. . . . . 6 |- ((F |` (C \ {A})) u. (/)) = (F |` (C \ {A}))
20	17, 18, 19	3eqtri 1506	. . . . 5 |- (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A}))) = (F |` (C \ {A}))
21	5, 6, 20	3eqtri 1506	. . . 4 |- (G |` (C \ {A})) = (F |` (C \ {A}))
22	21	fveq1i 3741	. . 3 |- ((G |` (C \ {A}))` D) = ((F |` (C \ {A}))` D)
23	2, 22	syl5eq 1526	. 2 |- (D e. (C \ {A}) -> ((G |` (C \ {A}))` D) = (F` D))
24	1, 23	eqtr3d 1516	1 |- (D e. (C \ {A}) -> (G` D) = (F` D))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 960   e. wcel 962  Vcvv 1818   \ cdif 2055   u. cun 2056   i^i cin 2057  (/)c0 2291  {csn 2421  <.cop 2423   |` cres 3188   Fn wfn 3193  -1-1-onto->wf1o 3197  ` cfv 3198

This theorem is referenced by:  facnn 6965  acdc2lem2 7522  acdc5lem2 7525  ruclem8 7550

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-sep 2718  ax-nul 2725  ax-pow 2758  ax-pr 2795

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-uni 2518  df-br 2635  df-opab 2682  df-id 2851  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fn 3209  df-f 3210  df-f1 3211  df-fo 3212  df-f1o 3213  df-fv 3214

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