Theorem cnmpt22f 13089

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
cnmpt21.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt21.a	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
cnmpt2t.b	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
cnmpt22f.f	⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))

Assertion

Ref	Expression
cnmpt22f	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt22f

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmpt21.j	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		cnmpt21.k	. 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		cnmpt21.a	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
4		cnmpt2t.b	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
5		cntop2 12996	. . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
6	3, 5	syl 14	. . 3 ⊢ (𝜑 → 𝐿 ∈ Top)
7		toptopon2 12811	. . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
8	6, 7	sylib 121	. 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
9		cntop2 12996	. . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
10	4, 9	syl 14	. . 3 ⊢ (𝜑 → 𝑀 ∈ Top)
11		toptopon2 12811	. . 3 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
12	10, 11	sylib 121	. 2 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
13		txtopon 13056	. . . . . . 7 ⊢ ((𝐿 ∈ (TopOn‘∪ 𝐿) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)))
14	8, 12, 13	syl2anc 409	. . . . . 6 ⊢ (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)))
15		cnmpt22f.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
16		cntop2 12996	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
17	15, 16	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Top)
18		toptopon2 12811	. . . . . . 7 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
19	17, 18	sylib 121	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
20		cnf2 12999	. . . . . 6 ⊢ (((𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)) ∧ 𝑁 ∈ (TopOn‘∪ 𝑁) ∧ 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁)
21	14, 19, 15, 20	syl3anc 1233	. . . . 5 ⊢ (𝜑 → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁)
22	21	ffnd 5348	. . . 4 ⊢ (𝜑 → 𝐹 Fn (∪ 𝐿 × ∪ 𝑀))
23		fnovim 5961	. . . 4 ⊢ (𝐹 Fn (∪ 𝐿 × ∪ 𝑀) → 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)))
24	22, 23	syl 14	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)))
25	24, 15	eqeltrrd 2248	. 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
26		oveq12 5862	. 2 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵))
27	1, 2, 3, 4, 8, 12, 25, 26	cnmpt22 13088	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  ∪ cuni 3796   × cxp 4609   Fn wfn 5193  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853   ∈ cmpo 5855  Topctop 12789  TopOnctopon 12802   Cn ccn 12979   ×_t ctx 13046

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982  df-tx 13047

This theorem is referenced by:  cnmptcom  13092  divcnap  13349  cnrehmeocntop  13387