Theorem off 6173

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 3393	. . . . . . 7 |- ( A i^i B ) C_ A
4	2, 3	eqsstrri 3226	. . . . . 6 |- C C_ A
5	4	sseli 3189	. . . . 5 |- ( z e. C -> z e. A )
6		ffvelcdm 5715	. . . . 5 |- ( ( F : A --> S /\ z e. A ) -> ( F ` z ) e. S )
7	1, 5, 6	syl2an 289	. . . 4 |- ( ( ph /\ z e. C ) -> ( F ` z ) e. S )
8		off.3	. . . . 5 |- ( ph -> G : B --> T )
9		inss2 3394	. . . . . . 7 |- ( A i^i B ) C_ B
10	2, 9	eqsstrri 3226	. . . . . 6 |- C C_ B
11	10	sseli 3189	. . . . 5 |- ( z e. C -> z e. B )
12		ffvelcdm 5715	. . . . 5 |- ( ( G : B --> T /\ z e. B ) -> ( G ` z ) e. T )
13	8, 11, 12	syl2an 289	. . . 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 2588	. . . . 5 |- ( ph -> A. x e. S A. y e. T ( x R y ) e. U )
16	15	adantr 276	. . . 4 |- ( ( ph /\ z e. C ) -> A. x e. S A. y e. T ( x R y ) e. U )
17		oveq1 5953	. . . . . 6 |- ( x = ( F ` z ) -> ( x R y ) = ( ( F ` z ) R y ) )
18	17	eleq1d 2274	. . . . 5 |- ( x = ( F ` z ) -> ( ( x R y ) e. U <-> ( ( F ` z ) R y ) e. U ) )
19		oveq2 5954	. . . . . 6 |- ( y = ( G ` z ) -> ( ( F ` z ) R y ) = ( ( F ` z ) R ( G ` z ) ) )
20	19	eleq1d 2274	. . . . 5 |- ( y = ( G ` z ) -> ( ( ( F ` z ) R y ) e. U <-> ( ( F ` z ) R ( G ` z ) ) e. U ) )
21	18, 20	rspc2va 2891	. . . 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 1249	. . 3 |- ( ( ph /\ z e. C ) -> ( ( F ` z ) R ( G ` z ) ) e. U )
23		eqid 2205	. . 3 |- ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) )
24	22, 23	fmptd 5736	. 2 |- ( ph -> ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) : C --> U )
25		ffn 5427	. . . . 5 |- ( F : A --> S -> F Fn A )
26	1, 25	syl 14	. . . 4 |- ( ph -> F Fn A )
27		ffn 5427	. . . . 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 2206	. . . 4 |- ( ( ph /\ z e. A ) -> ( F ` z ) = ( F ` z ) )
32		eqidd 2206	. . . 4 |- ( ( ph /\ z e. B ) -> ( G ` z ) = ( G ` z ) )
33	26, 28, 29, 30, 2, 31, 32	offval 6168	. . 3 |- ( ph -> ( F oF R G ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) )
34	33	feq1d 5414	. 2 |- ( ph -> ( ( F oF R G ) : C --> U <-> ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) : C --> U ) )
35	24, 34	mpbird 167	1 |- ( ph -> ( F oF R G ) : C --> U )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   i^i cin 3165   |-> cmpt 4106   Fn wfn 5267   -->wf 5268   ` cfv 5272  (class class class)co 5946   oFcof 6158

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-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-of 6160

This theorem is referenced by:  offeq  6174  ofnegsub  9037  lcomf  14122  psraddcl  14475  mplsubgfilemcl  14494  dvaddxxbr  15206  dvmulxxbr  15207  dvaddxx  15208  dvmulxx  15209  dviaddf  15210  dvimulf  15211  plyaddlem  15254