Theorem oprab2co 6427

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 4786	. . 3 |- ( ( C e. R /\ D e. S ) -> <. C , D >. e. ( R X. S ) )
4	1, 2, 3	syl2anc 411	. 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 6061	. . . . 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 6126	. . 3 |- ( x e. A , y e. B |-> ( C M D ) ) = ( x e. A , y e. B |-> ( M ` <. C , D >. ) )
10	6, 9	eqtri 2255	. 2 |- G = ( x e. A , y e. B |-> ( M ` <. C , D >. ) )
11	4, 5, 10	oprabco 6426	1 |- ( M Fn ( R X. S ) -> G = ( M o. F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   <.cop 3697   X. cxp 4752   o. ccom 4758   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   e. cmpo 6060

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348

This theorem is referenced by: (None)