Theorem cnmpt12 14730

Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-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))
cnmpt1t.b	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))
cnmpt12.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt12.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmpt12.c	⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
cnmpt12.d	⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
cnmpt12	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀))

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

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦,𝑧)   𝐷(𝑥)   𝐽(𝑧)   𝐾(𝑧)   𝐿(𝑧)

Proof of Theorem cnmpt12

Step	Hyp	Ref	Expression
1		cnmptid.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		cnmpt12.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		cnmpt11.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
4		cnf2 14648	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
5	1, 2, 3, 4	syl3anc 1249	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
6	5	fvmptelcdm 5732	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
7		cnmpt12.l	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
8		cnmpt1t.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿))
9		cnf2 14648	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
10	1, 7, 8, 9	syl3anc 1249	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍)
11	10	fvmptelcdm 5732	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑍)
12	6, 11	jca 306	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍))
13		txtopon 14705	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
14	2, 7, 13	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
15		cnmpt12.c	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
16		cntop2 14645	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top)
17	15, 16	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ Top)
18		toptopon2 14462	. . . . . . . . . 10 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
19	17, 18	sylib 122	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
20		cnf2 14648	. . . . . . . . 9 ⊢ (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀) ∧ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀)
21	14, 19, 15, 20	syl3anc 1249	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀)
22		eqid 2204	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) = (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)
23	22	fmpo 6286	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀)
24	21, 23	sylibr 134	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀)
25		r2al 2524	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀))
26	24, 25	sylib 122	. . . . . 6 ⊢ (𝜑 → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀))
27	26	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀))
28		eleq1 2267	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑌 ↔ 𝐴 ∈ 𝑌))
29		eleq1 2267	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝑍 ↔ 𝐵 ∈ 𝑍))
30	28, 29	bi2anan9 606	. . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍)))
31		cnmpt12.d	. . . . . . . 8 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → 𝐶 = 𝐷)
32	31	eleq1d 2273	. . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (𝐶 ∈ ∪ 𝑀 ↔ 𝐷 ∈ ∪ 𝑀))
33	30, 32	imbi12d 234	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀) ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 𝐷 ∈ ∪ 𝑀)))
34	33	spc2gv 2863	. . . . 5 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → (∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀) → ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 𝐷 ∈ ∪ 𝑀)))
35	12, 27, 12, 34	syl3c 63	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ∪ 𝑀)
36	31, 22	ovmpoga 6074	. . . 4 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍 ∧ 𝐷 ∈ ∪ 𝑀) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷)
37	6, 11, 35, 36	syl3anc 1249	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷)
38	37	mpteq2dva 4133	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐷))
39	1, 3, 8, 15	cnmpt12f 14729	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) ∈ (𝐽 Cn 𝑀))
40	38, 39	eqeltrrd 2282	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1370   = wceq 1372   ∈ wcel 2175  ∀wral 2483  ∪ cuni 3849   ↦ cmpt 4104   × cxp 4672  ⟶wf 5266  ‘cfv 5270  (class class class)co 5943   ∈ cmpo 5945  Topctop 14440  TopOnctopon 14453   Cn ccn 14628   ×_t ctx 14695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-map 6736  df-topgen 13063  df-top 14441  df-topon 14454  df-bases 14486  df-cn 14631  df-tx 14696

This theorem is referenced by:  plycn  15205