Theorem cnmpt11 23609

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 23195	. . . . . . . . . . 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 7058	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
8		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
9	8	fvmpt2 6952	. . . . . . . . 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 6838	. . . . . . 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 23187	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ Top)
18		toptopon2 22864	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
19	17, 18	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
20		cnf2 23195	. . . . . . . . . . . 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 7055	. . . . . . . . . . 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 3577	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ∪ 𝐿)
26	12, 13, 7, 25	fvmptd3 6964	. . . . . . 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 6933	. . . . . . 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 2736	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ 𝐶)
31	30	fvmpt2 6952	. . . . . . 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 2781	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥))
34	33	ralrimiva 3128	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥))
35		nfv 1915	. . . . 5 ⊢ Ⅎ𝑧(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐶)‘𝑥)
36		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌 ↦ 𝐵)
37		nfmpt1 5197	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ 𝐴)
38	36, 37	nfco 5814	. . . . . . 7 ⊢ Ⅎ𝑥((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))
39		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
40	38, 39	nffv 6844	. . . . . 6 ⊢ Ⅎ𝑥(((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧)
41		nfmpt1 5197	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ 𝐶)
42	41, 39	nffv 6844	. . . . . 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 6834	. . . . . 6 ⊢ (𝑥 = 𝑧 → (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑥) = (((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑧))
45		fveq2 6834	. . . . . 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 3278	. . . 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 6686	. . . . . 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 6663	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) Fn 𝑋)
52	25	fmpttd 7060	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶):𝑋⟶∪ 𝐿)
53	52	ffnd 6663	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) Fn 𝑋)
54		eqfnfv 6976	. . . 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 23212	. . 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 2837	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∪ cuni 4863   ↦ cmpt 5179   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Topctop 22839  TopOnctopon 22856   Cn ccn 23170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8767  df-top 22840  df-topon 22857  df-cn 23173

This theorem is referenced by:  cnmpt11f  23610  cnmptkp  23626  cnmptk1  23627  cnmpt1k  23628  ptunhmeo  23754  tmdgsum  24041  icchmeo  24896  icchmeoOLD  24897  evth2  24917  sinccvglem  35868  poimir  37856  broucube  37857