Theorem fovcld 6024

Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)

Hypothesis

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

Assertion

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

Proof of Theorem fovcld

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

Step	Hyp	Ref	Expression
1		3simpc 998	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> ( A e. R /\ B e. S ) )
2		fovcld.1	. . . 4 |- ( ph -> F : ( R X. S ) --> C )
3		ffnov 6023	. . . . 5 |- ( F : ( R X. S ) --> C <-> ( F Fn ( R X. S ) /\ A. x e. R A. y e. S ( x F y ) e. C ) )
4	3	simprbi 275	. . . 4 |- ( F : ( R X. S ) --> C -> A. x e. R A. y e. S ( x F y ) e. C )
5	2, 4	syl 14	. . 3 |- ( ph -> A. x e. R A. y e. S ( x F y ) e. C )
6	5	3ad2ant1 1020	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> A. x e. R A. y e. S ( x F y ) e. C )
7		oveq1 5926	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
8	7	eleq1d 2262	. . 3 |- ( x = A -> ( ( x F y ) e. C <-> ( A F y ) e. C ) )
9		oveq2 5927	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
10	9	eleq1d 2262	. . 3 |- ( y = B -> ( ( A F y ) e. C <-> ( A F B ) e. C ) )
11	8, 10	rspc2v 2878	. 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 ) )
12	1, 6, 11	sylc 62	1 |- ( ( ph /\ A e. R /\ B e. S ) -> ( A F B ) e. C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   X. cxp 4658   Fn wfn 5250   -->wf 5251  (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-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922

This theorem is referenced by:  fovcl  6025  imasrng  13455