Theorem fnbrfvb 3748

Description: Equivalence of function value and binary relation.

Hypothesis

Ref	Expression
fnfvbr.1	|- C e. V

Assertion

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

Proof of Theorem fnbrfvb

Step	Hyp	Ref	Expression
1		fnfvbr.1	. 2 |- C e. V
2		eqeq2 1482	. . . 4 |- (x = C -> ((F` B) = x <-> (F` B) = C))
3		breq2 2619	. . . 4 |- (x = C -> (BFx <-> BFC))
4	2, 3	bibi12d 628	. . 3 |- (x = C -> (((F` B) = x <-> BFx) <-> ((F` B) = C <-> BFC)))
5	4	imbi2d 611	. 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 3588	. . 3 |- ((F Fn A /\ B e. A) -> E!x BFx)
7		breq1 2618	. . . . . . 7 |- (y = B -> (yFx <-> BFx))
8	7	eubidv 1385	. . . . . 6 |- (y = B -> (E!x yFx <-> E!x BFx))
9		fveq2 3719	. . . . . . . 8 |- (y = B -> (F` y) = (F` B))
10	9	eqeq1d 1481	. . . . . . 7 |- (y = B -> ((F` y) = x <-> (F` B) = x))
11	10, 7	bibi12d 628	. . . . . 6 |- (y = B -> (((F` y) = x <-> yFx) <-> ((F` B) = x <-> BFx)))
12	8, 11	imbi12d 625	. . . . 5 |- (y = B -> ((E!x yFx -> ((F` y) = x <-> yFx)) <-> (E!x BFx -> ((F` B) = x <-> BFx))))
13		visset 1810	. . . . . 6 |- y e. V
14	13	tz6.12c 3735	. . . . 5 |- (E!x yFx -> ((F` y) = x <-> yFx))
15	12, 14	vtoclg 1844	. . . 4 |- (B e. A -> (E!x BFx -> ((F` B) = x <-> BFx)))
16	15	adantl 388	. . 3 |- ((F Fn A /\ B e. A) -> (E!x BFx -> ((F` B) = x <-> BFx)))
17	6, 16	mpd 26	. 2 |- ((F Fn A /\ B e. A) -> ((F` B) = x <-> BFx))
18	1, 5, 17	vtocl 1839	1 |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  E!weu 1379  Vcvv 1808   class class class wbr 2615   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  fnopfvb 3749  funbrfvb 3750  fnsnfv 3762  dffo4 3815  f1fv 3869  isomin 3894  isoini 3895  2ndconst 4090  adjbd1o 9974  bra11 9997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194

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