Theorem 2rbropap 5017

Description: Properties of a pair in a restricted binary relation M expressed as an ordered-pair class abstraction: M is the binary relation W restricted by the conditions ps and ta. (Contributed by AV, 31-Jan-2021.)

Hypotheses

Ref	Expression
2rbropap.1	|- ( ph -> M = { <. f , p >. | ( f W p /\ ps /\ ta ) } )
2rbropap.2	|- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )
2rbropap.3	|- ( ( f = F /\ p = P ) -> ( ta <-> th ) )

Assertion

Ref	Expression
2rbropap	|- ( ( ph /\ F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch /\ th ) ) )

Distinct variable groups:   f, F, p   P, f, p   f, W, p   ch, f, p   th, f, p

Allowed substitution hints:   ph( f, p)   ps( f, p)   ta( f, p)   M( f, p)   X( f, p)   Y( f, p)

Proof of Theorem 2rbropap

Step	Hyp	Ref	Expression
1		2rbropap.1	. . . 4 |- ( ph -> M = { <. f , p >. | ( f W p /\ ps /\ ta ) } )
2		3anass 1042	. . . . 5 |- ( ( f W p /\ ps /\ ta ) <-> ( f W p /\ ( ps /\ ta ) ) )
3	2	opabbii 4717	. . . 4 |- { <. f , p >. | ( f W p /\ ps /\ ta ) } = { <. f , p >. | ( f W p /\ ( ps /\ ta ) ) }
4	1, 3	syl6eq 2672	. . 3 |- ( ph -> M = { <. f , p >. | ( f W p /\ ( ps /\ ta ) ) } )
5		2rbropap.2	. . . 4 |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )
6		2rbropap.3	. . . 4 |- ( ( f = F /\ p = P ) -> ( ta <-> th ) )
7	5, 6	anbi12d 747	. . 3 |- ( ( f = F /\ p = P ) -> ( ( ps /\ ta ) <-> ( ch /\ th ) ) )
8	4, 7	rbropap 5016	. 2 |- ( ( ph /\ F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ( ch /\ th ) ) ) )
9		3anass 1042	. 2 |- ( ( F W P /\ ch /\ th ) <-> ( F W P /\ ( ch /\ th ) ) )
10	8, 9	syl6bbr 278	1 |- ( ( ph /\ F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch /\ th ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   {copab 4712

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-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-br 4654  df-opab 4713

This theorem is referenced by:  iswlkon  26553