Theorem funbrfv 3826

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 3264	. . . 4 |- ((Rel F /\ AFB) -> A e. V)
2		funrel 3608	. . . 4 |- (Fun F -> Rel F)
3	1, 2	sylan 450	. . 3 |- ((Fun F /\ AFB) -> A e. V)
4		funbrfv.1	. . . 4 |- B e. V
5		breq1 2672	. . . . . . 7 |- (x = A -> (xFy <-> AFy))
6	5	anbi2d 618	. . . . . 6 |- (x = A -> ((Fun F /\ xFy) <-> (Fun F /\ AFy)))
7		fveq2 3800	. . . . . . 7 |- (x = A -> (F` x) = (F` A))
8	7	eqeq1d 1520	. . . . . 6 |- (x = A -> ((F` x) = y <-> (F` A) = y))
9	6, 8	imbi12d 628	. . . . 5 |- (x = A -> (((Fun F /\ xFy) -> (F` x) = y) <-> ((Fun F /\ AFy) -> (F` A) = y)))
10		breq2 2673	. . . . . . 7 |- (y = B -> (AFy <-> AFB))
11	10	anbi2d 618	. . . . . 6 |- (y = B -> ((Fun F /\ AFy) <-> (Fun F /\ AFB)))
12		eqeq2 1521	. . . . . 6 |- (y = B -> ((F` A) = y <-> (F` A) = B))
13	11, 12	imbi12d 628	. . . . 5 |- (y = B -> (((Fun F /\ AFy) -> (F` A) = y) <-> ((Fun F /\ AFB) -> (F` A) = B)))
14		visset 1851	. . . . . . . 8 |- x e. V
15	14	tz6.12-1 3812	. . . . . . 7 |- ((xFy /\ E!y xFy) -> (F` x) = y)
16		funeu 3612	. . . . . . 7 |- ((Fun F /\ xFy) -> E!y xFy)
17	15, 16	sylan2 453	. . . . . 6 |- ((xFy /\ (Fun F /\ xFy)) -> (F` x) = y)
18	17	anabss7 505	. . . . 5 |- ((Fun F /\ xFy) -> (F` x) = y)
19	9, 13, 18	vtocl2g 1888	. . . 4 |- ((A e. V /\ B e. V) -> ((Fun F /\ AFB) -> (F` A) = B))
20	4, 19	mpan2 699	. . 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 371	1 |- (Fun F -> (AFB -> (F` A) = B))

Syntax hints:   -> wi 3   /\ wa 221   = wceq 988   e. wcel 990  E!weu 1413  Vcvv 1849   class class class wbr 2669  Rel wrel 3230  Fun wfun 3231  ` cfv 3237

This theorem is referenced by:  funopfv 3827  fvelima 3840  funiunfv 3942  cbvfo 3961

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-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fv 3253

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