Theorem sprmpt2 5911

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 5572	. . . . . 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 3816	. . . 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 457	. . 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 3864	. 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 5600	. 2 |- ( ( V e. _V /\ E e. _V ) -> { <. f , p >. | ( f ( V W E ) p /\ ps ) } e. _V )
15	2, 9, 10, 11, 14	ovmpt2d 5679	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 1285   e. wcel 1434   _Vcvv 2610   class class class wbr 3805   {copab 3858  (class class class)co 5563   |-> cmpt2 5565

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568

This theorem is referenced by: (None)