Theorem opropabco 15712

Description: Composition of an operator with a function abstraction.

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 = (BMC))}

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

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 4040	. . 3 |- ((B e. R /\ C e. S) -> <.B, C>. e. (R X. S))
4	1, 2, 3	syl11anc 524	. 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 = (BMC))}
7		df-opr 4886	. . . . . 6 |- (BMC) = (M` <.B, C>.)
8	7	eqeq2i 1894	. . . . 5 |- (y = (BMC) <-> y = (M` <.B, C>.))
9	8	anbi2i 538	. . . 4 |- ((x e. A /\ y = (BMC)) <-> (x e. A /\ y = (M` <.B, C>.)))
10	9	opabbii 3402	. . 3 |- {<.x, y>. | (x e. A /\ y = (BMC))} = {<.x, y>. | (x e. A /\ y = (M` <.B, C>.))}
11	6, 10	eqtri 1908	. 2 |- G = {<.x, y>. | (x e. A /\ y = (M` <.B, C>.))}
12	4, 5, 11	fnopabco 15711	1 |- (M Fn (R X. S) -> G = (M o. F))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  <.cop 3046  {copab 3395   X. cxp 3984   o. ccom 3990   Fn wfn 3993  ` cfv 3998  (class class class)co 4884

This theorem is referenced by:  cnopropabco 15917

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886

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