Theorem funbrfvbg 3773

Description: Function value in terms of a binary relation.

Assertion

Ref	Expression
funbrfvbg	|- ((Fun F /\ A e. dom F /\ B e. C) -> ((F` A) = B <-> AFB))

Proof of Theorem funbrfvbg

Step	Hyp	Ref	Expression
1		eqeq2 1491	. . . . . 6 |- (x = B -> ((F` A) = x <-> (F` A) = B))
2		breq2 2638	. . . . . 6 |- (x = B -> (AFx <-> AFB))
3	1, 2	bibi12d 632	. . . . 5 |- (x = B -> (((F` A) = x <-> AFx) <-> ((F` A) = B <-> AFB)))
4	3	imbi2d 615	. . . 4 |- (x = B -> (((Fun F /\ A e. dom F) -> ((F` A) = x <-> AFx)) <-> ((Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB))))
5		visset 1820	. . . . 5 |- x e. V
6	5	funbrfvb 3771	. . . 4 |- ((Fun F /\ A e. dom F) -> ((F` A) = x <-> AFx))
7	4, 6	vtoclg 1854	. . 3 |- (B e. C -> ((Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB)))
8	7	3impib 835	. 2 |- ((B e. C /\ Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB))
9	8	3coml 844	1 |- ((Fun F /\ A e. dom F /\ B e. C) -> ((F` A) = B <-> AFB))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 779   = wceq 960   e. wcel 962   class class class wbr 2634  dom cdm 3186  Fun wfun 3192  ` cfv 3198

This theorem is referenced by:  fvelimab 3781

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-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-rex 1657  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-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fn 3209  df-fv 3214

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