Theorem fnfvbr 2855

Description: Equivalence of function value and binary relation.

Hypothesis

Ref	Expression
fnfvbr.1	|- C e. V

Assertion

Ref	Expression
fnfvbr	|- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))

Proof of Theorem fnfvbr

Step	Hyp	Ref	Expression
1		fnfvbr.1	. 2 |- C e. V
2		cleq2 1110	. . . 4 |- (x = C -> ((F` B) = x <-> (F` B) = C))
3		breq2 2066	. . . 4 |- (x = C -> (BFx <-> BFC))
4	2, 3	bibi12d 477	. . 3 |- (x = C -> (((F` B) = x <-> BFx) <-> ((F` B) = C <-> BFC)))
5	4	imbi2d 464	. 2 |- (x = C -> (((F Fn A /\ B e. A) -> ((F` B) = x <-> BFx)) <-> ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))))
6		fneu 2728	. . 3 |- ((F Fn A /\ B e. A) -> E!x BFx)
7		breq1 2065	. . . . . . 7 |- (y = B -> (yFx <-> BFx))
8	7	bieudv 1013	. . . . . 6 |- (y = B -> (E!x yFx <-> E!x BFx))
9		fveq2 2832	. . . . . . . 8 |- (y = B -> (F` y) = (F` B))
10	9	cleq1d 1109	. . . . . . 7 |- (y = B -> ((F` y) = x <-> (F` B) = x))
11	10, 7	bibi12d 477	. . . . . 6 |- (y = B -> (((F` y) = x <-> yFx) <-> ((F` B) = x <-> BFx)))
12	8, 11	imbi12d 474	. . . . 5 |- (y = B -> ((E!x yFx -> ((F` y) = x <-> yFx)) <-> (E!x BFx -> ((F` B) = x <-> BFx))))
13		visset 1350	. . . . . 6 |- y e. V
14	13	tz6.12c 2846	. . . . 5 |- (E!x yFx -> ((F` y) = x <-> yFx))
15	12, 14	vtoclg 1383	. . . 4 |- (B e. A -> (E!x BFx -> ((F` B) = x <-> BFx)))
16	15	adantl 305	. . 3 |- ((F Fn A /\ B e. A) -> (E!x BFx -> ((F` B) = x <-> BFx)))
17	6, 16	mpd 46	. 2 |- ((F Fn A /\ B e. A) -> ((F` B) = x <-> BFx))
18	1, 5, 17	vtocl 1378	1 |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E!weu 1007   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  fnfvop 2856  fvelrn 2883  f1fv 2916  isomin 2937  isoini 2938

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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