Theorem brfvopab 6700

Description: The classes involved in a binary relation of a function value which is an ordered-pair class abstraction are sets. (Contributed by AV, 7-Jan-2021.)

Hypothesis

Ref	Expression
brfvopab.1	|- ( X e. _V -> ( F ` X ) = { <. y , z >. | ph } )

Assertion

Ref	Expression
brfvopab	|- ( A ( F ` X ) B -> ( X e. _V /\ A e. _V /\ B e. _V ) )

Proof of Theorem brfvopab

Step	Hyp	Ref	Expression
1		brfvopab.1	. . . . . . 7 |- ( X e. _V -> ( F ` X ) = { <. y , z >. | ph } )
2	1	breqd 4664	. . . . . 6 |- ( X e. _V -> ( A ( F ` X ) B <-> A { <. y , z >. | ph } B ) )
3		brabv 6699	. . . . . 6 |- ( A { <. y , z >. | ph } B -> ( A e. _V /\ B e. _V ) )
4	2, 3	syl6bi 243	. . . . 5 |- ( X e. _V -> ( A ( F ` X ) B -> ( A e. _V /\ B e. _V ) ) )
5	4	imdistani 726	. . . 4 |- ( ( X e. _V /\ A ( F ` X ) B ) -> ( X e. _V /\ ( A e. _V /\ B e. _V ) ) )
6		3anass 1042	. . . 4 |- ( ( X e. _V /\ A e. _V /\ B e. _V ) <-> ( X e. _V /\ ( A e. _V /\ B e. _V ) ) )
7	5, 6	sylibr 224	. . 3 |- ( ( X e. _V /\ A ( F ` X ) B ) -> ( X e. _V /\ A e. _V /\ B e. _V ) )
8	7	ex 450	. 2 |- ( X e. _V -> ( A ( F ` X ) B -> ( X e. _V /\ A e. _V /\ B e. _V ) ) )
9		fvprc 6185	. . 3 |- ( -. X e. _V -> ( F ` X ) = (/) )
10		breq 4655	. . . 4 |- ( ( F ` X ) = (/) -> ( A ( F ` X ) B <-> A (/) B ) )
11		br0 4701	. . . . 5 |- -. A (/) B
12	11	pm2.21i 116	. . . 4 |- ( A (/) B -> ( X e. _V /\ A e. _V /\ B e. _V ) )
13	10, 12	syl6bi 243	. . 3 |- ( ( F ` X ) = (/) -> ( A ( F ` X ) B -> ( X e. _V /\ A e. _V /\ B e. _V ) ) )
14	9, 13	syl 17	. 2 |- ( -. X e. _V -> ( A ( F ` X ) B -> ( X e. _V /\ A e. _V /\ B e. _V ) ) )
15	8, 14	pm2.61i 176	1 |- ( A ( F ` X ) B -> ( X e. _V /\ A e. _V /\ B e. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   {copab 4712   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896

This theorem is referenced by:  wlkprop  26507  wlkv  26508  isupwlkg  41718