Theorem oprabco 6864

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 466	. . 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 11	. . 3 |- ( H Fn D -> F = ( x e. A , y e. B |-> C ) )
5		dffn5 5911	. . . 4 |- ( H Fn D <-> H = ( z e. D |-> ( H ` z ) ) )
6	5	biimpi 194	. . 3 |- ( H Fn D -> H = ( z e. D |-> ( H ` z ) ) )
7		fveq2 5864	. . 3 |- ( z = C -> ( H ` z ) = ( H ` C ) )
8	2, 4, 6, 7	fmpt2co 6863	. 2 |- ( H Fn D -> ( H o. F ) = ( x e. A , y e. B |-> ( H ` C ) ) )
9		oprabco.3	. 2 |- G = ( x e. A , y e. B |-> ( H ` C ) )
10	8, 9	syl6reqr 2527	1 |- ( H Fn D -> G = ( H o. F ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   |-> cmpt 4505   o. ccom 5003   Fn wfn 5581   ` cfv 5586   |-> cmpt2 6284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782

This theorem is referenced by:  oprab2co  6865