Theorem oprabco 6404

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 277	. . 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 5713	. . 3 |- ( H Fn D -> H = ( z e. D |-> ( H ` z ) ) )
7		fveq2 5661	. . 3 |- ( z = C -> ( H ` z ) = ( H ` C ) )
8	3, 5, 6, 7	fmpoco 6403	. 2 |- ( H Fn D -> ( H o. F ) = ( x e. A , y e. B |-> ( H ` C ) ) )
9	1, 8	eqtr4id 2284	1 |- ( H Fn D -> G = ( H o. F ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   o. ccom 4744   Fn wfn 5338   ` cfv 5343   e. cmpo 6043

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-iun 3986  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-fv 5351  df-oprab 6045  df-mpo 6046  df-1st 6325  df-2nd 6326

This theorem is referenced by:  oprab2co  6405