Theorem offval2 5668

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

Hypotheses

Ref	Expression
offval2.1	|- (ph -> A e. V)
offval2.2	|- ((ph /\ x e. A) -> B e. W)
offval2.3	|- ((ph /\ x e. A) -> C e. X)
offval2.4	|- (ph -> F = (x e. A |-> B))
offval2.5	|- (ph -> G = (x e. A |-> C))

Assertion

Ref	Expression
offval2	|- (ph -> ( F o F R G) = (x e. A |-> (B R C)))

Distinct variable groups:   x,A   ph,x   x, R

Allowed substitution hints:   B(x)   C(x)   F(x)   G(x)   V(x)   W(x)   X(x)

Proof of Theorem offval2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval2.2	. . . . . 6 |- ((ph /\ x e. A) -> B e. W)
2	1	ralrimiva 2386	. . . . 5 |- (ph -> A.x e. A B e. W)
3		eqid 2037	. . . . . 6 |- (x e. A |-> B) = (x e. A |-> B)
4	3	fnmpt 4968	. . . . 5 |- (A.x e. A B e. W -> (x e. A |-> B) Fn A)
5	2, 4	syl 14	. . . 4 |- (ph -> (x e. A |-> B) Fn A)
6		offval2.4	. . . . 5 |- (ph -> F = (x e. A |-> B))
7	6	fneq1d 4932	. . . 4 |- (ph -> ( F Fn A <-> (x e. A |-> B) Fn A))
8	5, 7	mpbird 156	. . 3 |- (ph -> F Fn A)
9		offval2.3	. . . . . 6 |- ((ph /\ x e. A) -> C e. X)
10	9	ralrimiva 2386	. . . . 5 |- (ph -> A.x e. A C e. X)
11		eqid 2037	. . . . . 6 |- (x e. A |-> C) = (x e. A |-> C)
12	11	fnmpt 4968	. . . . 5 |- (A.x e. A C e. X -> (x e. A |-> C) Fn A)
13	10, 12	syl 14	. . . 4 |- (ph -> (x e. A |-> C) Fn A)
14		offval2.5	. . . . 5 |- (ph -> G = (x e. A |-> C))
15	14	fneq1d 4932	. . . 4 |- (ph -> ( G Fn A <-> (x e. A |-> C) Fn A))
16	13, 15	mpbird 156	. . 3 |- (ph -> G Fn A)
17		offval2.1	. . 3 |- (ph -> A e. V)
18		inidm 3140	. . 3 |- (A i^i A) = A
19	6	adantr 261	. . . 4 |- ((ph /\ y e. A) -> F = (x e. A |-> B))
20	19	fveq1d 5123	. . 3 |- ((ph /\ y e. A) -> ( F ` y) = ((x e. A |-> B) ` y))
21	14	adantr 261	. . . 4 |- ((ph /\ y e. A) -> G = (x e. A |-> C))
22	21	fveq1d 5123	. . 3 |- ((ph /\ y e. A) -> ( G ` y) = ((x e. A |-> C) ` y))
23	8, 16, 17, 17, 18, 20, 22	offval 5661	. 2 |- (ph -> ( F o F R G) = (y e. A |-> (((x e. A |-> B) ` y) R((x e. A |-> C) ` y))))
24		nffvmpt1 5129	. . . . 5 |- F/_x((x e. A |-> B) ` y)
25		nfcv 2175	. . . . 5 |- F/_x R
26		nffvmpt1 5129	. . . . 5 |- F/_x((x e. A |-> C) ` y)
27	24, 25, 26	nfov 5478	. . . 4 |- F/_x(((x e. A |-> B) ` y) R((x e. A |-> C) ` y))
28		nfcv 2175	. . . 4 |- F/_y(((x e. A |-> B) ` x) R((x e. A |-> C) ` x))
29		fveq2 5121	. . . . 5 |- (y = x -> ((x e. A |-> B) ` y) = ((x e. A |-> B) ` x))
30		fveq2 5121	. . . . 5 |- (y = x -> ((x e. A |-> C) ` y) = ((x e. A |-> C) ` x))
31	29, 30	oveq12d 5473	. . . 4 |- (y = x -> (((x e. A |-> B) ` y) R((x e. A |-> C) ` y)) = (((x e. A |-> B) ` x) R((x e. A |-> C) ` x)))
32	27, 28, 31	cbvmpt 3842	. . 3 |- (y e. A |-> (((x e. A |-> B) ` y) R((x e. A |-> C) ` y))) = (x e. A |-> (((x e. A |-> B) ` x) R((x e. A |-> C) ` x)))
33		simpr 103	. . . . . 6 |- ((ph /\ x e. A) -> x e. A)
34	3	fvmpt2 5197	. . . . . 6 |- ((x e. A /\ B e. W) -> ((x e. A |-> B) ` x) = B)
35	33, 1, 34	syl2anc 391	. . . . 5 |- ((ph /\ x e. A) -> ((x e. A |-> B) ` x) = B)
36	11	fvmpt2 5197	. . . . . 6 |- ((x e. A /\ C e. X) -> ((x e. A |-> C) ` x) = C)
37	33, 9, 36	syl2anc 391	. . . . 5 |- ((ph /\ x e. A) -> ((x e. A |-> C) ` x) = C)
38	35, 37	oveq12d 5473	. . . 4 |- ((ph /\ x e. A) -> (((x e. A |-> B) ` x) R((x e. A |-> C) ` x)) = (B R C))
39	38	mpteq2dva 3838	. . 3 |- (ph -> (x e. A |-> (((x e. A |-> B) ` x) R((x e. A |-> C) ` x))) = (x e. A |-> (B R C)))
40	32, 39	syl5eq 2081	. 2 |- (ph -> (y e. A |-> (((x e. A |-> B) ` y) R((x e. A |-> C) ` y))) = (x e. A |-> (B R C)))
41	23, 40	eqtrd 2069	1 |- (ph -> ( F o F R G) = (x e. A |-> (B R C)))

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   e. wcel 1390  A.wral 2300   |-> cmpt 3809   Fn wfn 4840   ` 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:  ofc12  5673  caofinvl  5675  caofcom  5676