Theorem cnmpt22f 23590

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 23156	. . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐿 ∈ Top)
7		toptopon2 22833	. . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
8	6, 7	sylib 218	. 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
9		cntop2 23156	. . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
10	4, 9	syl 17	. . 3 ⊢ (𝜑 → 𝑀 ∈ Top)
11		toptopon2 22833	. . 3 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
12	10, 11	sylib 218	. 2 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
13		txtopon 23506	. . . . . . 7 ⊢ ((𝐿 ∈ (TopOn‘∪ 𝐿) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)))
14	8, 12, 13	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)))
15		cnmpt22f.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
16		cntop2 23156	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
17	15, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Top)
18		toptopon2 22833	. . . . . . 7 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
19	17, 18	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
20		cnf2 23164	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁)
22	21	ffnd 6652	. . . 4 ⊢ (𝜑 → 𝐹 Fn (∪ 𝐿 × ∪ 𝑀))
23		fnov 7477	. . . 4 ⊢ (𝐹 Fn (∪ 𝐿 × ∪ 𝑀) ↔ 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)))
24	22, 23	sylib 218	. . 3 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)))
25	24, 15	eqeltrrd 2832	. 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
26		oveq12 7355	. 2 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵))
27	1, 2, 3, 4, 8, 12, 25, 26	cnmpt22 23589	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∪ cuni 4856   × cxp 5612   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Topctop 22808  TopOnctopon 22825   Cn ccn 23139   ×_t ctx 23475

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-topgen 17347  df-top 22809  df-topon 22826  df-bases 22861  df-cn 23142  df-tx 23477

This theorem is referenced by:  cnmptcom  23593  cnmpt2plusg  24003  istgp2  24006  cnmpt2vsca  24110  cnmpt2ds  24759  divcnOLD  24784  cnrehmeo  24878  cnrehmeoOLD  24879  htpycom  24902  htpyco1  24904  htpycc  24906  reparphti  24923  reparphtiOLD  24924  pcohtpylem  24946  cnmpt2ip  25175  cxpcnOLD  26682  vmcn  30679  dipcn  30700  mndpluscn  33939  cvxsconn  35287