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Theorem cnmpt11 21514
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 476 . . . . . . . . 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 21101 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
62, 3, 4, 5syl3anc 1366 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
7 eqid 2651 . . . . . . . . . . . 12 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
87fmpt 6421 . . . . . . . . . . 11 (∀𝑥𝑋 𝐴𝑌 ↔ (𝑥𝑋𝐴):𝑋𝑌)
96, 8sylibr 224 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
109r19.21bi 2961 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐴𝑌)
117fvmpt2 6330 . . . . . . . . 9 ((𝑥𝑋𝐴𝑌) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
121, 10, 11syl2anc 694 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
1312fveq2d 6233 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = ((𝑦𝑌𝐵)‘𝐴))
14 cnmpt11.b . . . . . . . . . . . . . 14 (𝜑 → (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿))
15 cntop2 21093 . . . . . . . . . . . . . 14 ((𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
1614, 15syl 17 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ Top)
17 eqid 2651 . . . . . . . . . . . . . 14 𝐿 = 𝐿
1817toptopon 20770 . . . . . . . . . . . . 13 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1916, 18sylib 208 . . . . . . . . . . . 12 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
20 cnf2 21101 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐵):𝑌 𝐿)
213, 19, 14, 20syl3anc 1366 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌𝐵):𝑌 𝐿)
22 eqid 2651 . . . . . . . . . . . 12 (𝑦𝑌𝐵) = (𝑦𝑌𝐵)
2322fmpt 6421 . . . . . . . . . . 11 (∀𝑦𝑌 𝐵 𝐿 ↔ (𝑦𝑌𝐵):𝑌 𝐿)
2421, 23sylibr 224 . . . . . . . . . 10 (𝜑 → ∀𝑦𝑌 𝐵 𝐿)
2524adantr 480 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 𝐿)
26 cnmpt11.c . . . . . . . . . . 11 (𝑦 = 𝐴𝐵 = 𝐶)
2726eleq1d 2715 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝐵 𝐿𝐶 𝐿))
2827rspcv 3336 . . . . . . . . 9 (𝐴𝑌 → (∀𝑦𝑌 𝐵 𝐿𝐶 𝐿))
2910, 25, 28sylc 65 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝐶 𝐿)
3026, 22fvmptg 6319 . . . . . . . 8 ((𝐴𝑌𝐶 𝐿) → ((𝑦𝑌𝐵)‘𝐴) = 𝐶)
3110, 29, 30syl2anc 694 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘𝐴) = 𝐶)
3213, 31eqtrd 2685 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = 𝐶)
33 fvco3 6314 . . . . . . 7 (((𝑥𝑋𝐴):𝑋𝑌𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
346, 33sylan 487 . . . . . 6 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
35 eqid 2651 . . . . . . . 8 (𝑥𝑋𝐶) = (𝑥𝑋𝐶)
3635fvmpt2 6330 . . . . . . 7 ((𝑥𝑋𝐶 𝐿) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
371, 29, 36syl2anc 694 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
3832, 34, 373eqtr4d 2695 . . . . 5 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
3938ralrimiva 2995 . . . 4 (𝜑 → ∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
40 nfv 1883 . . . . 5 𝑧(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥)
41 nfcv 2793 . . . . . . . 8 𝑥(𝑦𝑌𝐵)
42 nfmpt1 4780 . . . . . . . 8 𝑥(𝑥𝑋𝐴)
4341, 42nfco 5320 . . . . . . 7 𝑥((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))
44 nfcv 2793 . . . . . . 7 𝑥𝑧
4543, 44nffv 6236 . . . . . 6 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧)
46 nfmpt1 4780 . . . . . . 7 𝑥(𝑥𝑋𝐶)
4746, 44nffv 6236 . . . . . 6 𝑥((𝑥𝑋𝐶)‘𝑧)
4845, 47nfeq 2805 . . . . 5 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)
49 fveq2 6229 . . . . . 6 (𝑥 = 𝑧 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧))
50 fveq2 6229 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝑋𝐶)‘𝑥) = ((𝑥𝑋𝐶)‘𝑧))
5149, 50eqeq12d 2666 . . . . 5 (𝑥 = 𝑧 → ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5240, 48, 51cbvral 3197 . . . 4 (∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
5339, 52sylib 208 . . 3 (𝜑 → ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
54 fco 6096 . . . . . 6 (((𝑦𝑌𝐵):𝑌 𝐿 ∧ (𝑥𝑋𝐴):𝑋𝑌) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
5521, 6, 54syl2anc 694 . . . . 5 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
56 ffn 6083 . . . . 5 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋)
5755, 56syl 17 . . . 4 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋)
5829, 35fmptd 6425 . . . . 5 (𝜑 → (𝑥𝑋𝐶):𝑋 𝐿)
59 ffn 6083 . . . . 5 ((𝑥𝑋𝐶):𝑋 𝐿 → (𝑥𝑋𝐶) Fn 𝑋)
6058, 59syl 17 . . . 4 (𝜑 → (𝑥𝑋𝐶) Fn 𝑋)
61 eqfnfv 6351 . . . 4 ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋 ∧ (𝑥𝑋𝐶) Fn 𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
6257, 60, 61syl2anc 694 . . 3 (𝜑 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
6353, 62mpbird 247 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))
64 cnco 21118 . . 3 (((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
654, 14, 64syl2anc 694 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
6663, 65eqeltrrd 2731 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941   cuni 4468  cmpt 4762  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Topctop 20746  TopOnctopon 20763   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-top 20747  df-topon 20764  df-cn 21079
This theorem is referenced by:  cnmpt11f  21515  cnmptkp  21531  cnmptk1  21532  cnmpt1k  21533  ptunhmeo  21659  tmdgsum  21946  icchmeo  22787  evth2  22806  sinccvglem  31692  poimir  33572  broucube  33573
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