Theorem fnbrfvb 4712

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 1893	. . . 4 |- (x = C -> ((F` B) = x <-> (F` B) = C))
3		breq2 3342	. . . 4 |- (x = C -> (BFx <-> BFC))
4	2, 3	bibi12d 691	. . 3 |- (x = C -> (((F` B) = x <-> BFx) <-> ((F` B) = C <-> BFC)))
5	4	imbi2d 674	. 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 4517	. . 3 |- ((F Fn A /\ B e. A) -> E!x BFx)
7		breq1 3341	. . . . . . 7 |- (y = B -> (yFx <-> BFx))
8	7	eubidv 1779	. . . . . 6 |- (y = B -> (E!x yFx <-> E!x BFx))
9		fveq2 4681	. . . . . . . 8 |- (y = B -> (F` y) = (F` B))
10	9	eqeq1d 1892	. . . . . . 7 |- (y = B -> ((F` y) = x <-> (F` B) = x))
11	10, 7	bibi12d 691	. . . . . 6 |- (y = B -> (((F` y) = x <-> yFx) <-> ((F` B) = x <-> BFx)))
12	8, 11	imbi12d 688	. . . . 5 |- (y = B -> ((E!x yFx -> ((F` y) = x <-> yFx)) <-> (E!x BFx -> ((F` B) = x <-> BFx))))
13		visset 2295	. . . . . 6 |- y e. _V
14	13	tz6.12c 4697	. . . . 5 |- (E!x yFx -> ((F` y) = x <-> yFx))
15	12, 14	vtoclg 2346	. . . 4 |- (B e. A -> (E!x BFx -> ((F` B) = x <-> BFx)))
16	15	adantl 424	. . 3 |- ((F Fn A /\ B e. A) -> (E!x BFx -> ((F` B) = x <-> BFx)))
17	6, 16	mpd 29	. 2 |- ((F Fn A /\ B e. A) -> ((F` B) = x <-> BFx))
18	1, 5, 17	vtocl 2339	1 |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E!weu 1771  _Vcvv 2292   class class class wbr 3338   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  fnopfvb 4713  funbrfvb 4714  fnsnfv 4728  dffo4 4793  dff13 4850  isomin 4876  isoini 4877  1stconst 5070  2ndconst 5071  adjbd1o 11655  bra11 11679  trclss 13935  fvbigcup 14076

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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