Theorem fnbrfvb 3829

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 1521	. . . 4 |- (x = C -> ((F` B) = x <-> (F` B) = C))
3		breq2 2673	. . . 4 |- (x = C -> (BFx <-> BFC))
4	2, 3	bibi12d 631	. . 3 |- (x = C -> (((F` B) = x <-> BFx) <-> ((F` B) = C <-> BFC)))
5	4	imbi2d 614	. 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 3667	. . 3 |- ((F Fn A /\ B e. A) -> E!x BFx)
7		breq1 2672	. . . . . . 7 |- (y = B -> (yFx <-> BFx))
8	7	eubidv 1419	. . . . . 6 |- (y = B -> (E!x yFx <-> E!x BFx))
9		fveq2 3800	. . . . . . . 8 |- (y = B -> (F` y) = (F` B))
10	9	eqeq1d 1520	. . . . . . 7 |- (y = B -> ((F` y) = x <-> (F` B) = x))
11	10, 7	bibi12d 631	. . . . . 6 |- (y = B -> (((F` y) = x <-> yFx) <-> ((F` B) = x <-> BFx)))
12	8, 11	imbi12d 628	. . . . 5 |- (y = B -> ((E!x yFx -> ((F` y) = x <-> yFx)) <-> (E!x BFx -> ((F` B) = x <-> BFx))))
13		visset 1851	. . . . . 6 |- y e. V
14	13	tz6.12c 3816	. . . . 5 |- (E!x yFx -> ((F` y) = x <-> yFx))
15	12, 14	vtoclg 1885	. . . 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 1880	1 |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 988   e. wcel 990  E!weu 1413  Vcvv 1849   class class class wbr 2669   Fn wfn 3232  ` cfv 3237

This theorem is referenced by:  fnopfvb 3830  funbrfvb 3831  fnsnfv 3843  dffo4 3896  dff13 3950  isomin 3975  isoini 3976  1stconst 4206  2ndconst 4207  adjbd1o 10135  bra11 10158

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-fv 3253

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