Theorem sprmpt2 5880

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.)

Hypotheses

Ref	Expression
sprmpt2.1	|- M = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v W e ) p /\ ch ) } )
sprmpt2.2	|- ( ( v = V /\ e = E ) -> ( ch <-> ps ) )
sprmpt2.3	|- ( ( V e. _V /\ E e. _V ) -> ( f ( V W E ) p -> th ) )
sprmpt2.4	|- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | th } e. _V )

Assertion

Ref	Expression
sprmpt2	|- ( ( V e. _V /\ E e. _V ) -> ( V M E ) = { <. f , p >. | ( f ( V W E ) p /\ ps ) } )

Distinct variable groups:   e, E, f, p, v   e, V, f, p, v   e, W, v   ps, e, v

Allowed substitution hints:   ps( f, p)   ch( v, e, f, p)   th( v, e, f, p)   M( v, e, f, p)   W( f, p)

Proof of Theorem sprmpt2

Step	Hyp	Ref	Expression
1		sprmpt2.1	. . 3 |- M = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v W e ) p /\ ch ) } )
2	1	a1i 9	. 2 |- ( ( V e. _V /\ E e. _V ) -> M = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v W e ) p /\ ch ) } ) )
3		oveq12 5541	. . . . . 6 |- ( ( v = V /\ e = E ) -> ( v W e ) = ( V W E ) )
4	3	adantl 271	. . . . 5 |- ( ( ( V e. _V /\ E e. _V ) /\ ( v = V /\ e = E ) ) -> ( v W e ) = ( V W E ) )
5	4	breqd 3796	. . . 4 |- ( ( ( V e. _V /\ E e. _V ) /\ ( v = V /\ e = E ) ) -> ( f ( v W e ) p <-> f ( V W E ) p ) )
6		sprmpt2.2	. . . . 5 |- ( ( v = V /\ e = E ) -> ( ch <-> ps ) )
7	6	adantl 271	. . . 4 |- ( ( ( V e. _V /\ E e. _V ) /\ ( v = V /\ e = E ) ) -> ( ch <-> ps ) )
8	5, 7	anbi12d 456	. . 3 |- ( ( ( V e. _V /\ E e. _V ) /\ ( v = V /\ e = E ) ) -> ( ( f ( v W e ) p /\ ch ) <-> ( f ( V W E ) p /\ ps ) ) )
9	8	opabbidv 3844	. 2 |- ( ( ( V e. _V /\ E e. _V ) /\ ( v = V /\ e = E ) ) -> { <. f , p >. | ( f ( v W e ) p /\ ch ) } = { <. f , p >. | ( f ( V W E ) p /\ ps ) } )
10		simpl 107	. 2 |- ( ( V e. _V /\ E e. _V ) -> V e. _V )
11		simpr 108	. 2 |- ( ( V e. _V /\ E e. _V ) -> E e. _V )
12		sprmpt2.3	. . 3 |- ( ( V e. _V /\ E e. _V ) -> ( f ( V W E ) p -> th ) )
13		sprmpt2.4	. . 3 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | th } e. _V )
14	12, 13	opabbrex 5569	. 2 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V )
15	2, 9, 10, 11, 14	ovmpt2d 5648	1 |- ( ( V e. _V /\ E e. _V ) -> ( V M E ) = { <. f , p >. | ( f ( V W E ) p /\ ps ) } )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   {copab 3838  (class class class)co 5532   |-> cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537

This theorem is referenced by: (None)