Theorem cnmpt22f 13798

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 13705	. . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
6	3, 5	syl 14	. . 3 ⊢ (𝜑 → 𝐿 ∈ Top)
7		toptopon2 13522	. . 3 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
8	6, 7	sylib 122	. 2 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
9		cntop2 13705	. . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
10	4, 9	syl 14	. . 3 ⊢ (𝜑 → 𝑀 ∈ Top)
11		toptopon2 13522	. . 3 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
12	10, 11	sylib 122	. 2 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
13		txtopon 13765	. . . . . . 7 ⊢ ((𝐿 ∈ (TopOn‘∪ 𝐿) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)))
14	8, 12, 13	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(∪ 𝐿 × ∪ 𝑀)))
15		cnmpt22f.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
16		cntop2 13705	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁) → 𝑁 ∈ Top)
17	15, 16	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Top)
18		toptopon2 13522	. . . . . . 7 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
19	17, 18	sylib 122	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
20		cnf2 13708	. . . . . 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 1238	. . . . 5 ⊢ (𝜑 → 𝐹:(∪ 𝐿 × ∪ 𝑀)⟶∪ 𝑁)
22	21	ffnd 5367	. . . 4 ⊢ (𝜑 → 𝐹 Fn (∪ 𝐿 × ∪ 𝑀))
23		fnovim 5983	. . . 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 2255	. 2 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (𝑧𝐹𝑤)) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))
26		oveq12 5884	. 2 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → (𝑧𝐹𝑤) = (𝐴𝐹𝐵))
27	1, 2, 3, 4, 8, 12, 25, 26	cnmpt22 13797	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  ∪ cuni 3810   × cxp 4625   Fn wfn 5212  ⟶wf 5213  ‘cfv 5217  (class class class)co 5875   ∈ cmpo 5877  Topctop 13500  TopOnctopon 13513   Cn ccn 13688   ×_t ctx 13755

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-topgen 12709  df-top 13501  df-topon 13514  df-bases 13546  df-cn 13691  df-tx 13756

This theorem is referenced by:  cnmptcom  13801  divcnap  14058  cnrehmeocntop  14096