Theorem fovcl 5606

Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.)

Hypothesis

Ref	Expression
fovcl.1	|- F : ( R X. S ) --> C

Assertion

Ref	Expression
fovcl	|- ( ( A e. R /\ B e. S ) -> ( A F B ) e. C )

Proof of Theorem fovcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fovcl.1	. . 3 |- F : ( R X. S ) --> C
2		ffnov 5605	. . . 4 |- ( F : ( R X. S ) --> C <-> ( F Fn ( R X. S ) /\ A. x e. R A. y e. S ( x F y ) e. C ) )
3	2	simprbi 260	. . 3 |- ( F : ( R X. S ) --> C -> A. x e. R A. y e. S ( x F y ) e. C )
4	1, 3	ax-mp 7	. 2 |- A. x e. R A. y e. S ( x F y ) e. C
5		oveq1 5519	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
6	5	eleq1d 2106	. . 3 |- ( x = A -> ( ( x F y ) e. C <-> ( A F y ) e. C ) )
7		oveq2 5520	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
8	7	eleq1d 2106	. . 3 |- ( y = B -> ( ( A F y ) e. C <-> ( A F B ) e. C ) )
9	6, 8	rspc2v 2662	. 2 |- ( ( A e. R /\ B e. S ) -> ( A. x e. R A. y e. S ( x F y ) e. C -> ( A F B ) e. C ) )
10	4, 9	mpi 15	1 |- ( ( A e. R /\ B e. S ) -> ( A F B ) e. C )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   A.wral 2306   X. cxp 4343   Fn wfn 4897   -->wf 4898  (class class class)co 5512

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-ov 5515

This theorem is referenced by:  ixxssxr  8769  fzof  9001  elfzoelz  9004  fzoval  9005