Theorem cnmpt11 23601

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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
2		cnmptid.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		cnmpt11.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4		cnmpt11.a	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
5		cnf2 23187	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
6	2, 3, 4, 5	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
7	6	fvmptelcdm 7103	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
8		eqid 2735	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
9	8	fvmpt2 6997	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
10	1, 7, 9	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
11	10	fveq2d 6880	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = ((𝑦 ∈ 𝑌 ↦ 𝐵)‘𝐴))
12		eqid 2735	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑦 ∈ 𝑌 ↦ 𝐵)
13		cnmpt11.c	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝐵 = 𝐶)
14	13	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝐵 ∈ ∪ 𝐿 ↔ 𝐶 ∈ ∪ 𝐿))
15		cnmpt11.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿))
16		cntop2 23179	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ Top)
18		toptopon2 22856	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
19	17, 18	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
20		cnf2 23187	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿)
21	3, 19, 15, 20	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿)
22	12	fmpt 7100	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿)
23	21, 22	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝐿)
24	23	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝐿)
25	14, 24, 7	rspcdva 3602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ∪ 𝐿)
26	12, 13, 7, 25	fvmptd3 7009	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐵)‘𝐴) = 𝐶)
27	11, 26	eqtrd 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = 𝐶)
28		fvco3 6978	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌 ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)))
29	6, 28	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)))
30		eqid 2735	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ 𝐶)
31	30	fvmpt2 6997	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐶 ∈ ∪ 𝐿) → ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) = 𝐶)
32	1, 25, 31	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) = 𝐶)
33	27, 29, 32	3eqtr4d 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥))
34	33	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥))
35		nfv 1914	. . . . 5 ⊢ Ⅎ𝑧(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥)
36		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌 ↦ 𝐵)
37		nfmpt1 5220	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ 𝐴)
38	36, 37	nfco 5845	. . . . . . 7 ⊢ Ⅎ𝑥((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))
39		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
40	38, 39	nffv 6886	. . . . . 6 ⊢ Ⅎ𝑥(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧)
41		nfmpt1 5220	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ 𝐶)
42	41, 39	nffv 6886	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧)
43	40, 42	nfeq 2912	. . . . 5 ⊢ Ⅎ𝑥(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧)
44		fveq2 6876	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧))
45		fveq2 6876	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧))
46	44, 45	eqeq12d 2751	. . . . 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)))
47	35, 43, 46	cbvralw 3286	. . . 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))
48	34, 47	sylib 218	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧))
49		fco 6730	. . . . . 6 ⊢ (((𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿 ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)):𝑋⟶∪ 𝐿)
50	21, 6, 49	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)):𝑋⟶∪ 𝐿)
51	50	ffnd 6707	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) Fn 𝑋)
52	25	fmpttd 7105	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶):𝑋⟶∪ 𝐿)
53	52	ffnd 6707	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) Fn 𝑋)
54		eqfnfv 7021	. . . 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)))
55	51, 53, 54	syl2anc 584	. . 3 ⊢ (𝜑 → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶) ↔ ∀𝑧 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧)))
56	48, 55	mpbird 257	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
57		cnco 23204	. . 3 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿)) → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
58	4, 15, 57	syl2anc 584	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
59	56, 58	eqeltrrd 2835	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∪ cuni 4883   ↦ cmpt 5201   ∘ ccom 5658   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  Topctop 22831  TopOnctopon 22848   Cn ccn 23162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8842  df-top 22832  df-topon 22849  df-cn 23165

This theorem is referenced by:  cnmpt11f  23602  cnmptkp  23618  cnmptk1  23619  cnmpt1k  23620  ptunhmeo  23746  tmdgsum  24033  icchmeo  24889  icchmeoOLD  24890  evth2  24910  sinccvglem  35694  poimir  37677  broucube  37678