Theorem off 5666

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 Ry) 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 o F 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 3151	. . . . . . 7 |- (A i^i B) C_ A
4	2, 3	eqsstr3i 2970	. . . . . 6 |- C C_ A
5	4	sseli 2935	. . . . 5 |- (z e. C -> z e. A)
6		ffvelrn 5243	. . . . 5 |- (( F :A --> S /\ z e. A) -> ( F ` z) e. S)
7	1, 5, 6	syl2an 273	. . . 4 |- ((ph /\ z e. C) -> ( F ` z) e. S)
8		off.3	. . . . 5 |- (ph -> G :B --> T)
9		inss2 3152	. . . . . . 7 |- (A i^i B) C_ B
10	2, 9	eqsstr3i 2970	. . . . . 6 |- C C_ B
11	10	sseli 2935	. . . . 5 |- (z e. C -> z e. B)
12		ffvelrn 5243	. . . . 5 |- (( G :B --> T /\ z e. B) -> ( G ` z) e. T)
13	8, 11, 12	syl2an 273	. . . 4 |- ((ph /\ z e. C) -> ( G ` z) e. T)
14		off.1	. . . . . 6 |- ((ph /\ (x e. S /\ y e. T)) -> (x Ry) e. U)
15	14	ralrimivva 2395	. . . . 5 |- (ph -> A.x e. S A.y e. T (x Ry) e. U)
16	15	adantr 261	. . . 4 |- ((ph /\ z e. C) -> A.x e. S A.y e. T (x Ry) e. U)
17		oveq1 5462	. . . . . 6 |- (x = ( F ` z) -> (x Ry) = (( F ` z) Ry))
18	17	eleq1d 2103	. . . . 5 |- (x = ( F ` z) -> ((x Ry) e. U <-> (( F ` z) Ry) e. U))
19		oveq2 5463	. . . . . 6 |- (y = ( G ` z) -> (( F ` z) Ry) = (( F ` z) R( G ` z)))
20	19	eleq1d 2103	. . . . 5 |- (y = ( G ` z) -> ((( F ` z) Ry) e. U <-> (( F ` z) R( G ` z)) e. U))
21	18, 20	rspc2va 2657	. . . 4 |- (((( F ` z) e. S /\ ( G ` z) e. T) /\ A.x e. S A.y e. T (x Ry) e. U) -> (( F ` z) R( G ` z)) e. U)
22	7, 13, 16, 21	syl21anc 1133	. . 3 |- ((ph /\ z e. C) -> (( F ` z) R( G ` z)) e. U)
23		eqid 2037	. . 3 |- (z e. C |-> (( F ` z) R( G ` z))) = (z e. C |-> (( F ` z) R( G ` z)))
24	22, 23	fmptd 5265	. 2 |- (ph -> (z e. C |-> (( F ` z) R( G ` z))) : C --> U)
25		ffn 4989	. . . . 5 |- ( F :A --> S -> F Fn A)
26	1, 25	syl 14	. . . 4 |- (ph -> F Fn A)
27		ffn 4989	. . . . 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 2038	. . . 4 |- ((ph /\ z e. A) -> ( F ` z) = ( F ` z))
32		eqidd 2038	. . . 4 |- ((ph /\ z e. B) -> ( G ` z) = ( G ` z))
33	26, 28, 29, 30, 2, 31, 32	offval 5661	. . 3 |- (ph -> ( F o F R G) = (z e. C |-> (( F ` z) R( G ` z))))
34	33	feq1d 4977	. 2 |- (ph -> (( F o F R G) : C --> U <-> (z e. C |-> (( F ` z) R( G ` z))) : C --> U))
35	24, 34	mpbird 156	1 |- (ph -> ( F o F R G) : C --> U)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   e. wcel 1390  A.wral 2300   i^i cin 2910   |-> cmpt 3809   Fn wfn 4840   -->wf 4841   ` cfv 4845  (class class class)co 5455   o Fcof 5652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654

This theorem is referenced by: (None)