Theorem brfvopabrbr 6279

Description: The binary relation of a function value which is an ordered-pair class abstraction of a restricted binary relation is the restricted binary relation. The first hypothesis can often be obtained by using fvmptopab 6697. (Contributed by AV, 29-Oct-2021.)

Hypotheses

Ref	Expression
brfvopabrbr.1	|- ( A ` Z ) = { <. x , y >. | ( x ( B ` Z ) y /\ ph ) }
brfvopabrbr.2	|- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )
brfvopabrbr.3	|- Rel ( B ` Z )

Assertion

Ref	Expression
brfvopabrbr	|- ( X ( A ` Z ) Y <-> ( X ( B ` Z ) Y /\ ps ) )

Distinct variable groups:   x, B, y   x, X, y   x, Y, y   x, Z, y   ps, x, y

Allowed substitution hints:   ph( x, y)   A( x, y)

Proof of Theorem brfvopabrbr

Step	Hyp	Ref	Expression
1		brne0 4702	. . . 4 |- ( X ( A ` Z ) Y -> ( A ` Z ) =/= (/) )
2		fvprc 6185	. . . . 5 |- ( -. Z e. _V -> ( A ` Z ) = (/) )
3	2	necon1ai 2821	. . . 4 |- ( ( A ` Z ) =/= (/) -> Z e. _V )
4	1, 3	syl 17	. . 3 |- ( X ( A ` Z ) Y -> Z e. _V )
5		brfvopabrbr.1	. . . . 5 |- ( A ` Z ) = { <. x , y >. | ( x ( B ` Z ) y /\ ph ) }
6	5	relopabi 5245	. . . 4 |- Rel ( A ` Z )
7	6	brrelexi 5158	. . 3 |- ( X ( A ` Z ) Y -> X e. _V )
8	6	brrelex2i 5159	. . 3 |- ( X ( A ` Z ) Y -> Y e. _V )
9	4, 7, 8	3jca 1242	. 2 |- ( X ( A ` Z ) Y -> ( Z e. _V /\ X e. _V /\ Y e. _V ) )
10		brne0 4702	. . . . 5 |- ( X ( B ` Z ) Y -> ( B ` Z ) =/= (/) )
11		fvprc 6185	. . . . . 6 |- ( -. Z e. _V -> ( B ` Z ) = (/) )
12	11	necon1ai 2821	. . . . 5 |- ( ( B ` Z ) =/= (/) -> Z e. _V )
13	10, 12	syl 17	. . . 4 |- ( X ( B ` Z ) Y -> Z e. _V )
14		brfvopabrbr.3	. . . . 5 |- Rel ( B ` Z )
15	14	brrelexi 5158	. . . 4 |- ( X ( B ` Z ) Y -> X e. _V )
16	14	brrelex2i 5159	. . . 4 |- ( X ( B ` Z ) Y -> Y e. _V )
17	13, 15, 16	3jca 1242	. . 3 |- ( X ( B ` Z ) Y -> ( Z e. _V /\ X e. _V /\ Y e. _V ) )
18	17	adantr 481	. 2 |- ( ( X ( B ` Z ) Y /\ ps ) -> ( Z e. _V /\ X e. _V /\ Y e. _V ) )
19	5	a1i 11	. . 3 |- ( Z e. _V -> ( A ` Z ) = { <. x , y >. | ( x ( B ` Z ) y /\ ph ) } )
20		brfvopabrbr.2	. . 3 |- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )
21	19, 20	rbropap 5016	. 2 |- ( ( Z e. _V /\ X e. _V /\ Y e. _V ) -> ( X ( A ` Z ) Y <-> ( X ( B ` Z ) Y /\ ps ) ) )
22	9, 18, 21	pm5.21nii 368	1 |- ( X ( A ` Z ) Y <-> ( X ( B ` Z ) Y /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   {copab 4712   Rel wrel 5119   ` 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-xp 5120  df-rel 5121  df-iota 5851  df-fv 5896

This theorem is referenced by:  istrl  26593  ispth  26619  isspth  26620  isclwlk  26669  iscrct  26685  iscycl  26686  iseupth  27061