Theorem opabbrfexd 41307

Description: A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.)

Hypotheses

Ref	Expression
opabresexd.x	|- ( ( ph /\ x R y ) -> x e. C )
opabresexd.y	|- ( ( ph /\ x R y ) -> y : A --> B )
opabresexd.a	|- ( ( ph /\ x e. C ) -> A e. U )
opabresexd.b	|- ( ( ph /\ x e. C ) -> B e. V )
opabresexd.c	|- ( ph -> C e. W )

Assertion

Ref	Expression
opabbrfexd	|- ( ph -> { <. x , y >. | x R y } e. _V )

Distinct variable groups:   y, A   y, B   x, C, y   ph, x, y

Allowed substitution hints:   A( x)   B( x)   R( x, y)   U( x, y)   V( x, y)   W( x, y)

Proof of Theorem opabbrfexd

Step	Hyp	Ref	Expression
1		pm4.24 675	. . 3 |- ( x R y <-> ( x R y /\ x R y ) )
2	1	opabbii 4717	. 2 |- { <. x , y >. | x R y } = { <. x , y >. | ( x R y /\ x R y ) }
3		opabresexd.x	. . 3 |- ( ( ph /\ x R y ) -> x e. C )
4		opabresexd.y	. . 3 |- ( ( ph /\ x R y ) -> y : A --> B )
5		opabresexd.a	. . 3 |- ( ( ph /\ x e. C ) -> A e. U )
6		opabresexd.b	. . 3 |- ( ( ph /\ x e. C ) -> B e. V )
7		opabresexd.c	. . 3 |- ( ph -> C e. W )
8	3, 4, 5, 6, 7	opabresexd 41306	. 2 |- ( ph -> { <. x , y >. | ( x R y /\ x R y ) } e. _V )
9	2, 8	syl5eqel 2705	1 |- ( ph -> { <. x , y >. | x R y } e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   {copab 4712   -->wf 5884

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by: (None)