Theorem oprabco 6185

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.3	. 2 |- G = ( x e. A , y e. B |-> ( H ` C ) )
2		oprabco.1	. . . 4 |- ( ( x e. A /\ y e. B ) -> C e. D )
3	2	adantl 275	. . 3 |- ( ( H Fn D /\ ( x e. A /\ y e. B ) ) -> C e. D )
4		oprabco.2	. . . 4 |- F = ( x e. A , y e. B |-> C )
5	4	a1i 9	. . 3 |- ( H Fn D -> F = ( x e. A , y e. B |-> C ) )
6		dffn5im 5532	. . 3 |- ( H Fn D -> H = ( z e. D |-> ( H ` z ) ) )
7		fveq2 5486	. . 3 |- ( z = C -> ( H ` z ) = ( H ` C ) )
8	3, 5, 6, 7	fmpoco 6184	. 2 |- ( H Fn D -> ( H o. F ) = ( x e. A , y e. B |-> ( H ` C ) ) )
9	1, 8	eqtr4id 2218	1 |- ( H Fn D -> G = ( H o. F ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   o. ccom 4608   Fn wfn 5183   ` cfv 5188   e. cmpo 5844

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109

This theorem is referenced by:  oprab2co  6186