Theorem off 5744

Description: The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypotheses

Ref	Expression
off.1	|- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U )
off.2	|- ( ph -> F : A --> S )
off.3	|- ( ph -> G : B --> T )
off.4	|- ( ph -> A e. V )
off.5	|- ( ph -> B e. W )
off.6	|- ( A i^i B ) = C

Assertion

Ref	Expression
off	|- ( ph -> ( F oF R G ) : C --> U )

Distinct variable groups:   y, G   x, y, ph   x, S, y   x, T, y   x, F, y   x, R, y   x, U, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)   G( x)   V( x, y)   W( x, y)

Proof of Theorem off

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		off.2	. . . . 5 |- ( ph -> F : A --> S )
2		off.6	. . . . . . 7 |- ( A i^i B ) = C
3		inss1 3186	. . . . . . 7 |- ( A i^i B ) C_ A
4	2, 3	eqsstr3i 3030	. . . . . 6 |- C C_ A
5	4	sseli 2995	. . . . 5 |- ( z e. C -> z e. A )
6		ffvelrn 5321	. . . . 5 |- ( ( F : A --> S /\ z e. A ) -> ( F ` z ) e. S )
7	1, 5, 6	syl2an 283	. . . 4 |- ( ( ph /\ z e. C ) -> ( F ` z ) e. S )
8		off.3	. . . . 5 |- ( ph -> G : B --> T )
9		inss2 3187	. . . . . . 7 |- ( A i^i B ) C_ B
10	2, 9	eqsstr3i 3030	. . . . . 6 |- C C_ B
11	10	sseli 2995	. . . . 5 |- ( z e. C -> z e. B )
12		ffvelrn 5321	. . . . 5 |- ( ( G : B --> T /\ z e. B ) -> ( G ` z ) e. T )
13	8, 11, 12	syl2an 283	. . . 4 |- ( ( ph /\ z e. C ) -> ( G ` z ) e. T )
14		off.1	. . . . . 6 |- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U )
15	14	ralrimivva 2443	. . . . 5 |- ( ph -> A. x e. S A. y e. T ( x R y ) e. U )
16	15	adantr 270	. . . 4 |- ( ( ph /\ z e. C ) -> A. x e. S A. y e. T ( x R y ) e. U )
17		oveq1 5539	. . . . . 6 |- ( x = ( F ` z ) -> ( x R y ) = ( ( F ` z ) R y ) )
18	17	eleq1d 2147	. . . . 5 |- ( x = ( F ` z ) -> ( ( x R y ) e. U <-> ( ( F ` z ) R y ) e. U ) )
19		oveq2 5540	. . . . . 6 |- ( y = ( G ` z ) -> ( ( F ` z ) R y ) = ( ( F ` z ) R ( G ` z ) ) )
20	19	eleq1d 2147	. . . . 5 |- ( y = ( G ` z ) -> ( ( ( F ` z ) R y ) e. U <-> ( ( F ` z ) R ( G ` z ) ) e. U ) )
21	18, 20	rspc2va 2714	. . . 4 |- ( ( ( ( F ` z ) e. S /\ ( G ` z ) e. T ) /\ A. x e. S A. y e. T ( x R y ) e. U ) -> ( ( F ` z ) R ( G ` z ) ) e. U )
22	7, 13, 16, 21	syl21anc 1168	. . 3 |- ( ( ph /\ z e. C ) -> ( ( F ` z ) R ( G ` z ) ) e. U )
23		eqid 2081	. . 3 |- ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) )
24	22, 23	fmptd 5343	. 2 |- ( ph -> ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) : C --> U )
25		ffn 5066	. . . . 5 |- ( F : A --> S -> F Fn A )
26	1, 25	syl 14	. . . 4 |- ( ph -> F Fn A )
27		ffn 5066	. . . . 5 |- ( G : B --> T -> G Fn B )
28	8, 27	syl 14	. . . 4 |- ( ph -> G Fn B )
29		off.4	. . . 4 |- ( ph -> A e. V )
30		off.5	. . . 4 |- ( ph -> B e. W )
31		eqidd 2082	. . . 4 |- ( ( ph /\ z e. A ) -> ( F ` z ) = ( F ` z ) )
32		eqidd 2082	. . . 4 |- ( ( ph /\ z e. B ) -> ( G ` z ) = ( G ` z ) )
33	26, 28, 29, 30, 2, 31, 32	offval 5739	. . 3 |- ( ph -> ( F oF R G ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) )
34	33	feq1d 5054	. 2 |- ( ph -> ( ( F oF R G ) : C --> U <-> ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) : C --> U ) )
35	24, 34	mpbird 165	1 |- ( ph -> ( F oF R G ) : C --> U )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   i^i cin 2972   |-> cmpt 3839   Fn wfn 4917   -->wf 4918   ` cfv 4922  (class class class)co 5532   oFcof 5730

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-of 5732

This theorem is referenced by: (None)