Theorem off 6073

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 3347	. . . . . . 7 |- ( A i^i B ) C_ A
4	2, 3	eqsstrri 3180	. . . . . 6 |- C C_ A
5	4	sseli 3143	. . . . 5 |- ( z e. C -> z e. A )
6		ffvelrn 5629	. . . . 5 |- ( ( F : A --> S /\ z e. A ) -> ( F ` z ) e. S )
7	1, 5, 6	syl2an 287	. . . 4 |- ( ( ph /\ z e. C ) -> ( F ` z ) e. S )
8		off.3	. . . . 5 |- ( ph -> G : B --> T )
9		inss2 3348	. . . . . . 7 |- ( A i^i B ) C_ B
10	2, 9	eqsstrri 3180	. . . . . 6 |- C C_ B
11	10	sseli 3143	. . . . 5 |- ( z e. C -> z e. B )
12		ffvelrn 5629	. . . . 5 |- ( ( G : B --> T /\ z e. B ) -> ( G ` z ) e. T )
13	8, 11, 12	syl2an 287	. . . 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 2552	. . . . 5 |- ( ph -> A. x e. S A. y e. T ( x R y ) e. U )
16	15	adantr 274	. . . 4 |- ( ( ph /\ z e. C ) -> A. x e. S A. y e. T ( x R y ) e. U )
17		oveq1 5860	. . . . . 6 |- ( x = ( F ` z ) -> ( x R y ) = ( ( F ` z ) R y ) )
18	17	eleq1d 2239	. . . . 5 |- ( x = ( F ` z ) -> ( ( x R y ) e. U <-> ( ( F ` z ) R y ) e. U ) )
19		oveq2 5861	. . . . . 6 |- ( y = ( G ` z ) -> ( ( F ` z ) R y ) = ( ( F ` z ) R ( G ` z ) ) )
20	19	eleq1d 2239	. . . . 5 |- ( y = ( G ` z ) -> ( ( ( F ` z ) R y ) e. U <-> ( ( F ` z ) R ( G ` z ) ) e. U ) )
21	18, 20	rspc2va 2848	. . . 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 1232	. . 3 |- ( ( ph /\ z e. C ) -> ( ( F ` z ) R ( G ` z ) ) e. U )
23		eqid 2170	. . 3 |- ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) )
24	22, 23	fmptd 5650	. 2 |- ( ph -> ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) : C --> U )
25		ffn 5347	. . . . 5 |- ( F : A --> S -> F Fn A )
26	1, 25	syl 14	. . . 4 |- ( ph -> F Fn A )
27		ffn 5347	. . . . 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 2171	. . . 4 |- ( ( ph /\ z e. A ) -> ( F ` z ) = ( F ` z ) )
32		eqidd 2171	. . . 4 |- ( ( ph /\ z e. B ) -> ( G ` z ) = ( G ` z ) )
33	26, 28, 29, 30, 2, 31, 32	offval 6068	. . 3 |- ( ph -> ( F oF R G ) = ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) )
34	33	feq1d 5334	. 2 |- ( ph -> ( ( F oF R G ) : C --> U <-> ( z e. C |-> ( ( F ` z ) R ( G ` z ) ) ) : C --> U ) )
35	24, 34	mpbird 166	1 |- ( ph -> ( F oF R G ) : C --> U )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   i^i cin 3120   |-> cmpt 4050   Fn wfn 5193   -->wf 5194   ` cfv 5198  (class class class)co 5853   oFcof 6059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061

This theorem is referenced by:  offeq  6074  dvaddxxbr  13459  dvmulxxbr  13460  dvaddxx  13461  dvmulxx  13462  dviaddf  13463  dvimulf  13464