Theorem oprabco 6114

Description: Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Hypotheses

Ref	Expression
oprabco.1	|- ( ( x e. A /\ y e. B ) -> C e. D )
oprabco.2	|- F = ( x e. A , y e. B |-> C )
oprabco.3	|- G = ( x e. A , y e. B |-> ( H ` C ) )

Assertion

Ref	Expression
oprabco	|- ( H Fn D -> G = ( H o. F ) )

Distinct variable groups:   x, y, A   x, B, y   x, D, y   x, H, y

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

Proof of Theorem oprabco

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oprabco.1	. . . 4 |- ( ( x e. A /\ y e. B ) -> C e. D )
2	1	adantl 275	. . 3 |- ( ( H Fn D /\ ( x e. A /\ y e. B ) ) -> C e. D )
3		oprabco.2	. . . 4 |- F = ( x e. A , y e. B |-> C )
4	3	a1i 9	. . 3 |- ( H Fn D -> F = ( x e. A , y e. B |-> C ) )
5		dffn5im 5467	. . 3 |- ( H Fn D -> H = ( z e. D |-> ( H ` z ) ) )
6		fveq2 5421	. . 3 |- ( z = C -> ( H ` z ) = ( H ` C ) )
7	2, 4, 5, 6	fmpoco 6113	. 2 |- ( H Fn D -> ( H o. F ) = ( x e. A , y e. B |-> ( H ` C ) ) )
8		oprabco.3	. 2 |- G = ( x e. A , y e. B |-> ( H ` C ) )
9	7, 8	syl6reqr 2191	1 |- ( H Fn D -> G = ( H o. F ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   o. ccom 4543   Fn wfn 5118   ` cfv 5123   e. cmpo 5776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039

This theorem is referenced by:  oprab2co  6115