Theorem oprabco 6760

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 5839	. . . 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 5792	. . 3 |- ( z = C -> ( H ` z ) = ( H ` C ) )
8	2, 4, 6, 7	fmpt2co 6759	. 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 2511	1 |- ( H Fn D -> G = ( H o. F ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   |-> cmpt 4451   o. ccom 4945   Fn wfn 5514   ` cfv 5519   |-> cmpt2 6195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681

This theorem is referenced by:  oprab2co  6761