Theorem caovclg 6073

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 2576	. 2 |- ( ph -> A. x e. C A. y e. D ( x F y ) e. E )
3		oveq1 5926	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	eleq1d 2262	. . 3 |- ( x = A -> ( ( x F y ) e. E <-> ( A F y ) e. E ) )
5		oveq2 5927	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
6	5	eleq1d 2262	. . 3 |- ( y = B -> ( ( A F y ) e. E <-> ( A F B ) e. E ) )
7	4, 6	rspc2v 2878	. 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 1364   e. wcel 2164   A.wral 2472  (class class class)co 5919

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922

This theorem is referenced by:  caovcld  6074  caovcl  6075  caovlem2d  6113  frec2uzrdg  10483  frecuzrdgsuc  10488  iseqovex  10532  seq3val  10534  seqf  10538  seq3caopr  10569  seqcaoprg  10570  ercpbl  12917  grpinva  12972  imasgrp2  13183  imasrng  13455  imasring  13563  qusrhm  14027  qusmul2  14028  plymullem  14929