Theorem fnbrfvb 3738

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 1476	. . . 4 |- (x = C -> ((F` B) = x <-> (F` B) = C))
3		breq2 2613	. . . 4 |- (x = C -> (BFx <-> BFC))
4	2, 3	bibi12d 627	. . 3 |- (x = C -> (((F` B) = x <-> BFx) <-> ((F` B) = C <-> BFC)))
5	4	imbi2d 610	. 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 3578	. . 3 |- ((F Fn A /\ B e. A) -> E!x BFx)
7		breq1 2612	. . . . . . 7 |- (y = B -> (yFx <-> BFx))
8	7	eubidv 1379	. . . . . 6 |- (y = B -> (E!x yFx <-> E!x BFx))
9		fveq2 3709	. . . . . . . 8 |- (y = B -> (F` y) = (F` B))
10	9	eqeq1d 1475	. . . . . . 7 |- (y = B -> ((F` y) = x <-> (F` B) = x))
11	10, 7	bibi12d 627	. . . . . 6 |- (y = B -> (((F` y) = x <-> yFx) <-> ((F` B) = x <-> BFx)))
12	8, 11	imbi12d 624	. . . . 5 |- (y = B -> ((E!x yFx -> ((F` y) = x <-> yFx)) <-> (E!x BFx -> ((F` B) = x <-> BFx))))
13		visset 1804	. . . . . 6 |- y e. V
14	13	tz6.12c 3725	. . . . 5 |- (E!x yFx -> ((F` y) = x <-> yFx))
15	12, 14	vtoclg 1838	. . . 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 1833	1 |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E!weu 1373  Vcvv 1802   class class class wbr 2609   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  fnopfvb 3739  funbrfvb 3740  fnsnfv 3752  dffo4 3805  f1fv 3859  isomin 3884  isoini 3885  2ndconst 4081  adjbd1o 9933  bra11 9954

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-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

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