Theorem cnmpt11 23628

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 23214	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
6	2, 3, 4, 5	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
7	6	fvmptelcdm 7065	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
8		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
9	8	fvmpt2 6959	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
10	1, 7, 9	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
11	10	fveq2d 6844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = ((𝑦 ∈ 𝑌 ↦ 𝐵)‘𝐴))
12		eqid 2736	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑦 ∈ 𝑌 ↦ 𝐵)
13		cnmpt11.c	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝐵 = 𝐶)
14	13	eleq1d 2821	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝐵 ∈ ∪ 𝐿 ↔ 𝐶 ∈ ∪ 𝐿))
15		cnmpt11.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿))
16		cntop2 23206	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ Top)
18		toptopon2 22883	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
19	17, 18	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
20		cnf2 23214	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿)) → (𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿)
21	3, 19, 15, 20	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿)
22	12	fmpt 7062	. . . . . . . . . . 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 3565	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ∪ 𝐿)
26	12, 13, 7, 25	fvmptd3 6971	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐵)‘𝐴) = 𝐶)
27	11, 26	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)) = 𝐶)
28		fvco3 6939	. . . . . . 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 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑦 ∈ 𝑌 ↦ 𝐵)‘((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥)))
30		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ 𝐶)
31	30	fvmpt2 6959	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐶 ∈ ∪ 𝐿) → ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) = 𝐶)
32	1, 25, 31	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) = 𝐶)
33	27, 29, 32	3eqtr4d 2781	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥))
34	33	ralrimiva 3129	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥))
35		nfv 1916	. . . . 5 ⊢ Ⅎ𝑧(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥)
36		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌 ↦ 𝐵)
37		nfmpt1 5184	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ 𝐴)
38	36, 37	nfco 5820	. . . . . . 7 ⊢ Ⅎ𝑥((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))
39		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
40	38, 39	nffv 6850	. . . . . 6 ⊢ Ⅎ𝑥(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧)
41		nfmpt1 5184	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ 𝐶)
42	41, 39	nffv 6850	. . . . . 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 6840	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧))
45		fveq2 6840	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑧))
46	44, 45	eqeq12d 2752	. . . . 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 3279	. . . 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 6692	. . . . . 6 ⊢ (((𝑦 ∈ 𝑌 ↦ 𝐵):𝑌⟶∪ 𝐿 ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)):𝑋⟶∪ 𝐿)
50	21, 6, 49	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)):𝑋⟶∪ 𝐿)
51	50	ffnd 6669	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) Fn 𝑋)
52	25	fmpttd 7067	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶):𝑋⟶∪ 𝐿)
53	52	ffnd 6669	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) Fn 𝑋)
54		eqfnfv 6983	. . . 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 585	. . 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 23231	. . 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 585	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
59	56, 58	eqeltrrd 2837	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∪ cuni 4850   ↦ cmpt 5166   ∘ ccom 5635   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Topctop 22858  TopOnctopon 22875   Cn ccn 23189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775  df-top 22859  df-topon 22876  df-cn 23192

This theorem is referenced by:  cnmpt11f  23629  cnmptkp  23645  cnmptk1  23646  cnmpt1k  23647  ptunhmeo  23773  tmdgsum  24060  icchmeo  24908  evth2  24927  sinccvglem  35854  poimir  37974  broucube  37975