Theorem fnbrfvb 3753

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 1484	. . . 4 |- (x = C -> ((F` B) = x <-> (F` B) = C))
3		breq2 2623	. . . 4 |- (x = C -> (BFx <-> BFC))
4	2, 3	bibi12d 629	. . 3 |- (x = C -> (((F` B) = x <-> BFx) <-> ((F` B) = C <-> BFC)))
5	4	imbi2d 612	. 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 3592	. . 3 |- ((F Fn A /\ B e. A) -> E!x BFx)
7		breq1 2622	. . . . . . 7 |- (y = B -> (yFx <-> BFx))
8	7	eubidv 1386	. . . . . 6 |- (y = B -> (E!x yFx <-> E!x BFx))
9		fveq2 3724	. . . . . . . 8 |- (y = B -> (F` y) = (F` B))
10	9	eqeq1d 1483	. . . . . . 7 |- (y = B -> ((F` y) = x <-> (F` B) = x))
11	10, 7	bibi12d 629	. . . . . 6 |- (y = B -> (((F` y) = x <-> yFx) <-> ((F` B) = x <-> BFx)))
12	8, 11	imbi12d 626	. . . . 5 |- (y = B -> ((E!x yFx -> ((F` y) = x <-> yFx)) <-> (E!x BFx -> ((F` B) = x <-> BFx))))
13		visset 1813	. . . . . 6 |- y e. V
14	13	tz6.12c 3740	. . . . 5 |- (E!x yFx -> ((F` y) = x <-> yFx))
15	12, 14	vtoclg 1847	. . . 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 1842	1 |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E!weu 1380  Vcvv 1811   class class class wbr 2619   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  fnopfvb 3754  funbrfvb 3755  fnsnfv 3767  dffo4 3820  f1fv 3874  isomin 3899  isoini 3900  2ndconst 4097  adjbd1o 10018  bra11 10041

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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