Theorem oprab2co 6454

Description: Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)

Hypotheses

Ref	Expression
oprab2co.1	|- ( ( x e. A /\ y e. B ) -> C e. R )
oprab2co.2	|- ( ( x e. A /\ y e. B ) -> D e. S )
oprab2co.3	|- F = ( x e. A , y e. B |-> <. C , D >. )
oprab2co.4	|- G = ( x e. A , y e. B |-> ( C M D ) )

Assertion

Ref	Expression
oprab2co	|- ( M Fn ( R X. S ) -> G = ( M o. F ) )

Distinct variable groups:   x, y, A   x, B, y   x, M, y   x, R, y   x, S, y

Allowed substitution hints:   C( x, y)   D( x, y)   F( x, y)   G( x, y)

Proof of Theorem oprab2co

Step	Hyp	Ref	Expression
1		oprab2co.1	. . 3 |- ( ( x e. A /\ y e. B ) -> C e. R )
2		oprab2co.2	. . 3 |- ( ( x e. A /\ y e. B ) -> D e. S )
3		opelxpi 4806	. . 3 |- ( ( C e. R /\ D e. S ) -> <. C , D >. e. ( R X. S ) )
4	1, 2, 3	syl2anc 415	. 2 |- ( ( x e. A /\ y e. B ) -> <. C , D >. e. ( R X. S ) )
5		oprab2co.3	. 2 |- F = ( x e. A , y e. B |-> <. C , D >. )
6		oprab2co.4	. . 3 |- G = ( x e. A , y e. B |-> ( C M D ) )
7		df-ov 6088	. . . . 5 |- ( C M D ) = ( M ` <. C , D >. )
8	7	a1i 9	. . . 4 |- ( ( x e. A /\ y e. B ) -> ( C M D ) = ( M ` <. C , D >. ) )
9	8	mpoeq3ia 6153	. . 3 |- ( x e. A , y e. B |-> ( C M D ) ) = ( x e. A , y e. B |-> ( M ` <. C , D >. ) )
10	6, 9	eqtri 2259	. 2 |- G = ( x e. A , y e. B |-> ( M ` <. C , D >. ) )
11	4, 5, 10	oprabco 6453	1 |- ( M Fn ( R X. S ) -> G = ( M o. F ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   <.cop 3712   X. cxp 4772   o. ccom 4778   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   e. cmpo 6087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375

This theorem is used by: (None)