Theorem funbrfvbg 3742

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 1476	. . . . . 6 |- (x = B -> ((F` A) = x <-> (F` A) = B))
2		breq2 2613	. . . . . 6 |- (x = B -> (AFx <-> AFB))
3	1, 2	bibi12d 627	. . . . 5 |- (x = B -> (((F` A) = x <-> AFx) <-> ((F` A) = B <-> AFB)))
4	3	imbi2d 610	. . . 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 1804	. . . . 5 |- x e. V
6	5	funbrfvb 3740	. . . 4 |- ((Fun F /\ A e. dom F) -> ((F` A) = x <-> AFx))
7	4, 6	vtoclg 1838	. . 3 |- (B e. C -> ((Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB)))
8	7	3impib 829	. 2 |- ((B e. C /\ Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB))
9	8	3coml 838	1 |- ((Fun F /\ A e. dom F /\ B e. C) -> ((F` A) = B <-> AFB))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   class class class wbr 2609  dom cdm 3160  Fun wfun 3166  ` cfv 3172

This theorem is referenced by:  fvelimab 3750

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

Copyright terms: Public domain