Theorem caovclg 6360

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 2908	. 2 |- ( ph -> A. x e. C A. y e. D ( x F y ) e. E )
3		oveq1 6202	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	eleq1d 2521	. . 3 |- ( x = A -> ( ( x F y ) e. E <-> ( A F y ) e. E ) )
5		oveq2 6203	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
6	5	eleq1d 2521	. . 3 |- ( y = B -> ( ( A F y ) e. E <-> ( A F B ) e. E ) )
7	4, 6	rspc2v 3180	. 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 469	1 |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2796  (class class class)co 6195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-iota 5484  df-fv 5529  df-ov 6198

This theorem is referenced by:  caovcld  6361  caovcl  6362  grprinvd  6407  seqcl2  11936  seqcaopr  11955  ercpbl  14601  imasmnd2  15572  gsumpropd2lem  15619  imasgrp2  15784  gsumzaddlem  16524  gsumzaddlemOLD  16526  imasrng  16829  divsrhm  17437  mplind  17703  plymullem  21812