Theorem caovclg 5931

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 2517	. 2 |- ( ph -> A. x e. C A. y e. D ( x F y ) e. E )
3		oveq1 5789	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
4	3	eleq1d 2209	. . 3 |- ( x = A -> ( ( x F y ) e. E <-> ( A F y ) e. E ) )
5		oveq2 5790	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
6	5	eleq1d 2209	. . 3 |- ( y = B -> ( ( A F y ) e. E <-> ( A F B ) e. E ) )
7	4, 6	rspc2v 2806	. 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 279	1 |- ( ( ph /\ ( A e. C /\ B e. D ) ) -> ( A F B ) e. E )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   A.wral 2417  (class class class)co 5782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  caovcld  5932  caovcl  5933  caovlem2d  5971  grprinvd  5974  frec2uzrdg  10213  frecuzrdgsuc  10218  iseqovex  10260  seq3val  10262  seqf  10265  seq3caopr  10287