Theorem opropabco 33518

Description: Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem opropabco

Step	Hyp	Ref	Expression
1		opropabco.1	. . 3 |- ( x e. A -> B e. R )
2		opropabco.2	. . 3 |- ( x e. A -> C e. S )
3		opelxpi 5148	. . 3 |- ( ( B e. R /\ C e. S ) -> <. B , C >. e. ( R X. S ) )
4	1, 2, 3	syl2anc 693	. 2 |- ( x e. A -> <. B , C >. e. ( R X. S ) )
5		opropabco.3	. 2 |- F = { <. x , y >. | ( x e. A /\ y = <. B , C >. ) }
6		opropabco.4	. . 3 |- G = { <. x , y >. | ( x e. A /\ y = ( B M C ) ) }
7		df-ov 6653	. . . . . 6 |- ( B M C ) = ( M ` <. B , C >. )
8	7	eqeq2i 2634	. . . . 5 |- ( y = ( B M C ) <-> y = ( M ` <. B , C >. ) )
9	8	anbi2i 730	. . . 4 |- ( ( x e. A /\ y = ( B M C ) ) <-> ( x e. A /\ y = ( M ` <. B , C >. ) ) )
10	9	opabbii 4717	. . 3 |- { <. x , y >. | ( x e. A /\ y = ( B M C ) ) } = { <. x , y >. | ( x e. A /\ y = ( M ` <. B , C >. ) ) }
11	6, 10	eqtri 2644	. 2 |- G = { <. x , y >. | ( x e. A /\ y = ( M ` <. B , C >. ) ) }
12	4, 5, 11	fnopabco 33517	1 |- ( M Fn ( R X. S ) -> G = ( M o. F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   {copab 4712   X. cxp 5112   o. ccom 5118   Fn wfn 5883   ` cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)