Theorem caovclg 6158

Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)

Hypothesis

Ref	Expression
caovclg.1	|- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E )

Assertion

Ref	Expression
caovclg	|- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )

Distinct variable groups:   x, y, A   y, B   x, C, y   x, D, y   x, E, y   ph, x, y   x, F, y

Allowed substitution hint:   B( x)

Proof of Theorem caovclg

Step	Hyp	Ref	Expression
1		caovclg.1	. . 3 |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x F y ) e. E )
2	1	ralrimivva 2612	. 2 |- ( ph -> A. x e. C A. y e. D ( x F y ) e. E )
3		oveq1 6008	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	eleq1d 2298	. . 3 |- ( x = A -> ( ( x F y ) e. E <-> ( A F y ) e. E ) )
5		oveq2 6009	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
6	5	eleq1d 2298	. . 3 |- ( y = B -> ( ( A F y ) e. E <-> ( A F B ) e. E ) )
7	4, 6	rspc2v 2920	. 2 |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ( x F y ) e. E -> ( A F B ) e. E ) )
8	2, 7	mpan9 281	1 |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508  (class class class)co 6001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  caovcld  6159  caovcl  6160  caovlem2d  6198  frec2uzrdg  10631  frecuzrdgsuc  10636  iseqovex  10680  seq3val  10682  seqf  10686  seq3caopr  10717  seqcaoprg  10718  ercpbl  13364  grpinva  13419  imasmnd2  13485  imasgrp2  13647  imasrng  13919  imasring  14027  qusrhm  14492  qusmul2  14493  plymullem  15424