Theorem off 6243

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 3425	. . . . . . 7 |- ( A i^i B ) C_ A
4	2, 3	eqsstrri 3258	. . . . . 6 |- C C_ A
5	4	sseli 3221	. . . . 5 |- ( z e. C -> z e. A )
6		ffvelcdm 5776	. . . . 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 3426	. . . . . . 7 |- ( A i^i B ) C_ B
10	2, 9	eqsstrri 3258	. . . . . 6 |- C C_ B
11	10	sseli 3221	. . . . 5 |- ( z e. C -> z e. B )
12		ffvelcdm 5776	. . . . 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 2612	. . . . 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 6020	. . . . . 6 |- ( x = ( F ` z ) -> ( x R y ) = ( ( F ` z ) R y ) )
18	17	eleq1d 2298	. . . . 5 |- ( x = ( F ` z ) -> ( ( x R y ) e. U <-> ( ( F ` z ) R y ) e. U ) )
19		oveq2 6021	. . . . . 6 |- ( y = ( G ` z ) -> ( ( F ` z ) R y ) = ( ( F ` z ) R ( G ` z ) ) )
20	19	eleq1d 2298	. . . . 5 |- ( y = ( G ` z ) -> ( ( ( F ` z ) R y ) e. U <-> ( ( F ` z ) R ( G ` z ) ) e. U ) )
21	18, 20	rspc2va 2922	. . . 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 1270	. . 3 |- ( ( ph /\ z e. C ) -> ( ( F ` z ) R ( G ` z ) ) e. U )
23		eqid 2229	. . 3 |- ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) )
24	22, 23	fmptd 5797	. 2 |- ( ph -> ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) : C --> U )
25		ffn 5479	. . . . 5 |- ( F : A --> S -> F Fn A )
26	1, 25	syl 14	. . . 4 |- ( ph -> F Fn A )
27		ffn 5479	. . . . 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 2230	. . . 4 |- ( ( ph /\ z e. A ) -> ( F ` z ) = ( F ` z ) )
32		eqidd 2230	. . . 4 |- ( ( ph /\ z e. B ) -> ( G ` z ) = ( G ` z ) )
33	26, 28, 29, 30, 2, 31, 32	offval 6238	. . 3 |- ( ph -> ( F oF R G ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) )
34	33	feq1d 5466	. 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 1395   e. wcel 2200   A.wral 2508   i^i cin 3197   |-> cmpt 4148   Fn wfn 5319   -->wf 5320   ` cfv 5324  (class class class)co 6013   oFcof 6228

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230

This theorem is referenced by:  offeq  6244  ofnegsub  9132  lcomf  14331  psraddcl  14684  mplsubgfilemcl  14703  dvaddxxbr  15415  dvmulxxbr  15416  dvaddxx  15417  dvmulxx  15418  dviaddf  15419  dvimulf  15420  plyaddlem  15463