Theorem funbrfvbg 3763

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 1487	. . . . . 6 |- (x = B -> ((F` A) = x <-> (F` A) = B))
2		breq2 2628	. . . . . 6 |- (x = B -> (AFx <-> AFB))
3	1, 2	bibi12d 631	. . . . 5 |- (x = B -> (((F` A) = x <-> AFx) <-> ((F` A) = B <-> AFB)))
4	3	imbi2d 614	. . . 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 1816	. . . . 5 |- x e. V
6	5	funbrfvb 3761	. . . 4 |- ((Fun F /\ A e. dom F) -> ((F` A) = x <-> AFx))
7	4, 6	vtoclg 1850	. . 3 |- (B e. C -> ((Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB)))
8	7	3impib 833	. 2 |- ((B e. C /\ Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB))
9	8	3coml 842	1 |- ((Fun F /\ A e. dom F /\ B e. C) -> ((F` A) = B <-> AFB))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   class class class wbr 2624  dom cdm 3176  Fun wfun 3182  ` cfv 3188

This theorem is referenced by:  fvelimab 3771

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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