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Theorem cnmpt11 12643
 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 (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt11.a (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt11.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt11.b (𝜑 → (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿))
cnmpt11.c (𝑦 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
cnmpt11 (𝜑 → (𝑥𝑋𝐶) ∈ (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.
StepHypRef Expression
1 simpr 109 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝑥𝑋)
2 cnmptid.j . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 cnmpt11.k . . . . . . . . . . . 12 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 cnmpt11.a . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
5 cnf2 12565 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
62, 3, 4, 5syl3anc 1220 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
7 eqid 2157 . . . . . . . . . . . 12 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
87fmpt 5614 . . . . . . . . . . 11 (∀𝑥𝑋 𝐴𝑌 ↔ (𝑥𝑋𝐴):𝑋𝑌)
96, 8sylibr 133 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
109r19.21bi 2545 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐴𝑌)
117fvmpt2 5548 . . . . . . . . 9 ((𝑥𝑋𝐴𝑌) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
121, 10, 11syl2anc 409 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
1312fveq2d 5469 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = ((𝑦𝑌𝐵)‘𝐴))
14 eqid 2157 . . . . . . . 8 (𝑦𝑌𝐵) = (𝑦𝑌𝐵)
15 cnmpt11.c . . . . . . . 8 (𝑦 = 𝐴𝐵 = 𝐶)
1615eleq1d 2226 . . . . . . . . 9 (𝑦 = 𝐴 → (𝐵 𝐿𝐶 𝐿))
17 cnmpt11.b . . . . . . . . . . . . . 14 (𝜑 → (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿))
18 cntop2 12562 . . . . . . . . . . . . . 14 ((𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
1917, 18syl 14 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ Top)
20 eqid 2157 . . . . . . . . . . . . . 14 𝐿 = 𝐿
2120toptopon 12376 . . . . . . . . . . . . 13 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2219, 21sylib 121 . . . . . . . . . . . 12 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
23 cnf2 12565 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐵):𝑌 𝐿)
243, 22, 17, 23syl3anc 1220 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌𝐵):𝑌 𝐿)
2514fmpt 5614 . . . . . . . . . . 11 (∀𝑦𝑌 𝐵 𝐿 ↔ (𝑦𝑌𝐵):𝑌 𝐿)
2624, 25sylibr 133 . . . . . . . . . 10 (𝜑 → ∀𝑦𝑌 𝐵 𝐿)
2726adantr 274 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 𝐿)
2816, 27, 10rspcdva 2821 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝐶 𝐿)
2914, 15, 10, 28fvmptd3 5558 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘𝐴) = 𝐶)
3013, 29eqtrd 2190 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = 𝐶)
31 fvco3 5536 . . . . . . 7 (((𝑥𝑋𝐴):𝑋𝑌𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
326, 31sylan 281 . . . . . 6 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
33 eqid 2157 . . . . . . . 8 (𝑥𝑋𝐶) = (𝑥𝑋𝐶)
3433fvmpt2 5548 . . . . . . 7 ((𝑥𝑋𝐶 𝐿) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
351, 28, 34syl2anc 409 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
3630, 32, 353eqtr4d 2200 . . . . 5 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
3736ralrimiva 2530 . . . 4 (𝜑 → ∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
38 nfv 1508 . . . . 5 𝑧(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥)
39 nfcv 2299 . . . . . . . 8 𝑥(𝑦𝑌𝐵)
40 nfmpt1 4057 . . . . . . . 8 𝑥(𝑥𝑋𝐴)
4139, 40nfco 4748 . . . . . . 7 𝑥((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))
42 nfcv 2299 . . . . . . 7 𝑥𝑧
4341, 42nffv 5475 . . . . . 6 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧)
44 nfmpt1 4057 . . . . . . 7 𝑥(𝑥𝑋𝐶)
4544, 42nffv 5475 . . . . . 6 𝑥((𝑥𝑋𝐶)‘𝑧)
4643, 45nfeq 2307 . . . . 5 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)
47 fveq2 5465 . . . . . 6 (𝑥 = 𝑧 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧))
48 fveq2 5465 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝑋𝐶)‘𝑥) = ((𝑥𝑋𝐶)‘𝑧))
4947, 48eqeq12d 2172 . . . . 5 (𝑥 = 𝑧 → ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5038, 46, 49cbvral 2676 . . . 4 (∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
5137, 50sylib 121 . . 3 (𝜑 → ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
52 fco 5332 . . . . . 6 (((𝑦𝑌𝐵):𝑌 𝐿 ∧ (𝑥𝑋𝐴):𝑋𝑌) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
5324, 6, 52syl2anc 409 . . . . 5 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
5453ffnd 5317 . . . 4 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋)
5528fmpttd 5619 . . . . 5 (𝜑 → (𝑥𝑋𝐶):𝑋 𝐿)
5655ffnd 5317 . . . 4 (𝜑 → (𝑥𝑋𝐶) Fn 𝑋)
57 eqfnfv 5562 . . . 4 ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋 ∧ (𝑥𝑋𝐶) Fn 𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5854, 56, 57syl2anc 409 . . 3 (𝜑 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5951, 58mpbird 166 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))
60 cnco 12581 . . 3 (((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
614, 17, 60syl2anc 409 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
6259, 61eqeltrrd 2235 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335   ∈ wcel 2128  ∀wral 2435  ∪ cuni 3772   ↦ cmpt 4025   ∘ ccom 4587   Fn wfn 5162  ⟶wf 5163  ‘cfv 5167  (class class class)co 5818  Topctop 12355  TopOnctopon 12368   Cn ccn 12545 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-map 6588  df-top 12356  df-topon 12369  df-cn 12548 This theorem is referenced by:  cnmpt11f  12644
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