Theorem funbrfv 3750

Description: The second argument of a binary relation on a function is the function's value.

Hypothesis

Ref	Expression
funbrfv.1	|- B e. V

Assertion

Ref	Expression
funbrfv	|- (Fun F -> (AFB -> (F` A) = B))

Proof of Theorem funbrfv

Step	Hyp	Ref	Expression
1		brrelex 3207	. . . 4 |- ((Rel F /\ AFB) -> A e. V)
2		funrel 3533	. . . 4 |- (Fun F -> Rel F)
3	1, 2	sylan 448	. . 3 |- ((Fun F /\ AFB) -> A e. V)
4		funbrfv.1	. . . 4 |- B e. V
5		breq1 2622	. . . . . . 7 |- (x = A -> (xFy <-> AFy))
6	5	anbi2d 616	. . . . . 6 |- (x = A -> ((Fun F /\ xFy) <-> (Fun F /\ AFy)))
7		fveq2 3724	. . . . . . 7 |- (x = A -> (F` x) = (F` A))
8	7	eqeq1d 1483	. . . . . 6 |- (x = A -> ((F` x) = y <-> (F` A) = y))
9	6, 8	imbi12d 626	. . . . 5 |- (x = A -> (((Fun F /\ xFy) -> (F` x) = y) <-> ((Fun F /\ AFy) -> (F` A) = y)))
10		breq2 2623	. . . . . . 7 |- (y = B -> (AFy <-> AFB))
11	10	anbi2d 616	. . . . . 6 |- (y = B -> ((Fun F /\ AFy) <-> (Fun F /\ AFB)))
12		eqeq2 1484	. . . . . 6 |- (y = B -> ((F` A) = y <-> (F` A) = B))
13	11, 12	imbi12d 626	. . . . 5 |- (y = B -> (((Fun F /\ AFy) -> (F` A) = y) <-> ((Fun F /\ AFB) -> (F` A) = B)))
14		visset 1813	. . . . . . . 8 |- x e. V
15	14	tz6.12-1 3736	. . . . . . 7 |- ((xFy /\ E!y xFy) -> (F` x) = y)
16		funeu 3537	. . . . . . 7 |- ((Fun F /\ xFy) -> E!y xFy)
17	15, 16	sylan2 451	. . . . . 6 |- ((xFy /\ (Fun F /\ xFy)) -> (F` x) = y)
18	17	anabss7 503	. . . . 5 |- ((Fun F /\ xFy) -> (F` x) = y)
19	9, 13, 18	vtocl2g 1850	. . . 4 |- ((A e. V /\ B e. V) -> ((Fun F /\ AFB) -> (F` A) = B))
20	4, 19	mpan2 696	. . 3 |- (A e. V -> ((Fun F /\ AFB) -> (F` A) = B))
21	3, 20	mpcom 49	. 2 |- ((Fun F /\ AFB) -> (F` A) = B)
22	21	ex 373	1 |- (Fun F -> (AFB -> (F` A) = B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E!weu 1380  Vcvv 1811   class class class wbr 2619  Rel wrel 3175  Fun wfun 3176  ` cfv 3182

This theorem is referenced by:  funopfv 3751  fvelima 3764  funiunfv 3866  cbvfo 3885

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-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198

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