Theorem cnmpt11 12643

Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmptid.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt11.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt11.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt11.b	⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿))
cnmpt11.c	⊢ (𝑦 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cnmpt11	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜑,𝑥   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝐵   𝑦,𝐶

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)

Proof of Theorem cnmpt11

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 109	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
2		cnmptid.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		cnmpt11.k	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4		cnmpt11.a	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
5		cnf2 12565	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
6	2, 3, 4, 5	syl3anc 1220	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
7		eqid 2157	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
8	7	fmpt 5614	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌 ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
9	6, 8	sylibr 133	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌)
10	9	r19.21bi 2545	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
11	7	fvmpt2 5548	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
12	1, 10, 11	syl2anc 409	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
13	12	fveq2d 5469	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = ((𝑦 ∈ 𝑌 ↦ 𝐵)‘𝐴))
14		eqid 2157	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑦 ∈ 𝑌 ↦ 𝐵)
15		cnmpt11.c	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝐵 = 𝐶)
16	15	eleq1d 2226	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝐵 ∈ ∪ 𝐿 ↔ 𝐶 ∈ ∪ 𝐿))
17		cnmpt11.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿))
18		cntop2 12562	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
19	17, 18	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ Top)
20		eqid 2157	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐿 = ∪ 𝐿
21	20	toptopon 12376	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
22	19, 21	sylib 121	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
23		cnf2 12565	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿)
24	3, 22, 17, 23	syl3anc 1220	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿)
25	14	fmpt 5614	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿)
26	24, 25	sylibr 133	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝐿)
27	26	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝐿)
28	16, 27, 10	rspcdva 2821	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ∪ 𝐿)
29	14, 15, 10, 28	fvmptd3 5558	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐵)‘𝐴) = 𝐶)
30	13, 29	eqtrd 2190	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = 𝐶)
31		fvco3 5536	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌 ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)))
32	6, 31	sylan 281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)))
33		eqid 2157	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ 𝐶)
34	33	fvmpt2 5548	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐶 ∈ ∪ 𝐿) → ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) = 𝐶)
35	1, 28, 34	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) = 𝐶)
36	30, 32, 35	3eqtr4d 2200	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥))
37	36	ralrimiva 2530	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥))
38		nfv 1508	. . . . 5 ⊢ Ⅎ𝑧(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥)
39		nfcv 2299	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌 ↦ 𝐵)
40		nfmpt1 4057	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ 𝐴)
41	39, 40	nfco 4748	. . . . . . 7 ⊢ Ⅎ𝑥((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))
42		nfcv 2299	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
43	41, 42	nffv 5475	. . . . . 6 ⊢ Ⅎ𝑥(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧)
44		nfmpt1 4057	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ 𝐶)
45	44, 42	nffv 5475	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧)
46	43, 45	nfeq 2307	. . . . 5 ⊢ Ⅎ𝑥(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧)
47		fveq2 5465	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧))
48		fveq2 5465	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧))
49	47, 48	eqeq12d 2172	. . . . 5 ⊢ (𝑥 = 𝑧 → ((((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) ↔ (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧)))
50	38, 46, 49	cbvral 2676	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) ↔ ∀𝑧 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧))
51	37, 50	sylib 121	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧))
52		fco 5332	. . . . . 6 ⊢ (((𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿 ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)):𝑋⟶∪ 𝐿)
53	24, 6, 52	syl2anc 409	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)):𝑋⟶∪ 𝐿)
54	53	ffnd 5317	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) Fn 𝑋)
55	28	fmpttd 5619	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶):𝑋⟶∪ 𝐿)
56	55	ffnd 5317	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) Fn 𝑋)
57		eqfnfv 5562	. . . 4 ⊢ ((((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) Fn 𝑋 ∧ (𝑥 ∈ 𝑋 ↦ 𝐶) Fn 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶) ↔ ∀𝑧 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧)))
58	54, 56, 57	syl2anc 409	. . 3 ⊢ (𝜑 → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶) ↔ ∀𝑧 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧)))
59	51, 58	mpbird 166	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
60		cnco 12581	. . 3 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿)) → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
61	4, 17, 60	syl2anc 409	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
62	59, 61	eqeltrrd 2235	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335   ∈ wcel 2128  ∀wral 2435  ∪ cuni 3772   ↦ cmpt 4025   ∘ ccom 4587   Fn wfn 5162  ⟶wf 5163  ‘cfv 5167  (class class class)co 5818  Topctop 12355  TopOnctopon 12368   Cn ccn 12545

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-map 6588  df-top 12356  df-topon 12369  df-cn 12548

This theorem is referenced by:  cnmpt11f  12644