Theorem sprmpt2d 7350

Description: The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 20-Jun-2019.)

Hypotheses

Ref	Expression
sprmpt2d.1	|- M = ( v e. _V , e e. _V |-> { <. x , y >. | ( x ( v R e ) y /\ ch ) } )
sprmpt2d.2	|- ( ( ph /\ v = V /\ e = E ) -> ( ch <-> ps ) )
sprmpt2d.3	|- ( ph -> ( V e. _V /\ E e. _V ) )
sprmpt2d.4	|- ( ph -> A. x A. y ( x ( V R E ) y -> th ) )
sprmpt2d.5	|- ( ph -> { <. x , y >. | th } e. _V )

Assertion

Ref	Expression
sprmpt2d	|- ( ph -> ( V M E ) = { <. x , y >. | ( x ( V R E ) y /\ ps ) } )

Distinct variable groups:   e, E, v, x, y   R, e, v   e, V, v, x, y   ph, e, v, x, y   ps, e, v

Allowed substitution hints:   ps( x, y)   ch( x, y, v, e)   th( x, y, v, e)   R( x, y)   M( x, y, v, e)

Proof of Theorem sprmpt2d

Step	Hyp	Ref	Expression
1		sprmpt2d.1	. . 3 |- M = ( v e. _V , e e. _V |-> { <. x , y >. | ( x ( v R e ) y /\ ch ) } )
2	1	a1i 11	. 2 |- ( ph -> M = ( v e. _V , e e. _V |-> { <. x , y >. | ( x ( v R e ) y /\ ch ) } ) )
3		oveq12 6659	. . . . . 6 |- ( ( v = V /\ e = E ) -> ( v R e ) = ( V R E ) )
4	3	breqd 4664	. . . . 5 |- ( ( v = V /\ e = E ) -> ( x ( v R e ) y <-> x ( V R E ) y ) )
5	4	adantl 482	. . . 4 |- ( ( ph /\ ( v = V /\ e = E ) ) -> ( x ( v R e ) y <-> x ( V R E ) y ) )
6		sprmpt2d.2	. . . . 5 |- ( ( ph /\ v = V /\ e = E ) -> ( ch <-> ps ) )
7	6	3expb 1266	. . . 4 |- ( ( ph /\ ( v = V /\ e = E ) ) -> ( ch <-> ps ) )
8	5, 7	anbi12d 747	. . 3 |- ( ( ph /\ ( v = V /\ e = E ) ) -> ( ( x ( v R e ) y /\ ch ) <-> ( x ( V R E ) y /\ ps ) ) )
9	8	opabbidv 4716	. 2 |- ( ( ph /\ ( v = V /\ e = E ) ) -> { <. x , y >. | ( x ( v R e ) y /\ ch ) } = { <. x , y >. | ( x ( V R E ) y /\ ps ) } )
10		sprmpt2d.3	. . 3 |- ( ph -> ( V e. _V /\ E e. _V ) )
11	10	simpld 475	. 2 |- ( ph -> V e. _V )
12	10	simprd 479	. 2 |- ( ph -> E e. _V )
13		sprmpt2d.4	. . 3 |- ( ph -> A. x A. y ( x ( V R E ) y -> th ) )
14		sprmpt2d.5	. . 3 |- ( ph -> { <. x , y >. | th } e. _V )
15		opabbrex 6695	. . 3 |- ( ( A. x A. y ( x ( V R E ) y -> th ) /\ { <. x , y >. | th } e. _V ) -> { <. x , y >. | ( x ( V R E ) y /\ ps ) } e. _V )
16	13, 14, 15	syl2anc 693	. 2 |- ( ph -> { <. x , y >. | ( x ( V R E ) y /\ ps ) } e. _V )
17	2, 9, 11, 12, 16	ovmpt2d 6788	1 |- ( ph -> ( V M E ) = { <. x , y >. | ( x ( V R E ) y /\ ps ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   {copab 4712  (class class class)co 6650   |-> cmpt2 6652

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by: (None)