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Theorem cnmpt11 22894
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 485 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝑥𝑋)
2 cnmptid.j . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 cnmpt11.k . . . . . . . . . . 11 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 cnmpt11.a . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
5 cnf2 22480 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
62, 3, 4, 5syl3anc 1370 . . . . . . . . . 10 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
76fvmptelcdm 7026 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐴𝑌)
8 eqid 2736 . . . . . . . . . 10 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
98fvmpt2 6925 . . . . . . . . 9 ((𝑥𝑋𝐴𝑌) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
101, 7, 9syl2anc 584 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
1110fveq2d 6815 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = ((𝑦𝑌𝐵)‘𝐴))
12 eqid 2736 . . . . . . . 8 (𝑦𝑌𝐵) = (𝑦𝑌𝐵)
13 cnmpt11.c . . . . . . . 8 (𝑦 = 𝐴𝐵 = 𝐶)
1413eleq1d 2821 . . . . . . . . 9 (𝑦 = 𝐴 → (𝐵 𝐿𝐶 𝐿))
15 cnmpt11.b . . . . . . . . . . . . . 14 (𝜑 → (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿))
16 cntop2 22472 . . . . . . . . . . . . . 14 ((𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
1715, 16syl 17 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ Top)
18 toptopon2 22147 . . . . . . . . . . . . 13 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1917, 18sylib 217 . . . . . . . . . . . 12 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
20 cnf2 22480 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐵):𝑌 𝐿)
213, 19, 15, 20syl3anc 1370 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌𝐵):𝑌 𝐿)
2212fmpt 7023 . . . . . . . . . . 11 (∀𝑦𝑌 𝐵 𝐿 ↔ (𝑦𝑌𝐵):𝑌 𝐿)
2321, 22sylibr 233 . . . . . . . . . 10 (𝜑 → ∀𝑦𝑌 𝐵 𝐿)
2423adantr 481 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 𝐿)
2514, 24, 7rspcdva 3570 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝐶 𝐿)
2612, 13, 7, 25fvmptd3 6937 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘𝐴) = 𝐶)
2711, 26eqtrd 2776 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = 𝐶)
28 fvco3 6906 . . . . . . 7 (((𝑥𝑋𝐴):𝑋𝑌𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
296, 28sylan 580 . . . . . 6 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
30 eqid 2736 . . . . . . . 8 (𝑥𝑋𝐶) = (𝑥𝑋𝐶)
3130fvmpt2 6925 . . . . . . 7 ((𝑥𝑋𝐶 𝐿) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
321, 25, 31syl2anc 584 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
3327, 29, 323eqtr4d 2786 . . . . 5 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
3433ralrimiva 3139 . . . 4 (𝜑 → ∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
35 nfv 1916 . . . . 5 𝑧(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥)
36 nfcv 2904 . . . . . . . 8 𝑥(𝑦𝑌𝐵)
37 nfmpt1 5194 . . . . . . . 8 𝑥(𝑥𝑋𝐴)
3836, 37nfco 5794 . . . . . . 7 𝑥((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))
39 nfcv 2904 . . . . . . 7 𝑥𝑧
4038, 39nffv 6821 . . . . . 6 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧)
41 nfmpt1 5194 . . . . . . 7 𝑥(𝑥𝑋𝐶)
4241, 39nffv 6821 . . . . . 6 𝑥((𝑥𝑋𝐶)‘𝑧)
4340, 42nfeq 2917 . . . . 5 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)
44 fveq2 6811 . . . . . 6 (𝑥 = 𝑧 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧))
45 fveq2 6811 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝑋𝐶)‘𝑥) = ((𝑥𝑋𝐶)‘𝑧))
4644, 45eqeq12d 2752 . . . . 5 (𝑥 = 𝑧 → ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
4735, 43, 46cbvralw 3285 . . . 4 (∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
4834, 47sylib 217 . . 3 (𝜑 → ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
49 fco 6661 . . . . . 6 (((𝑦𝑌𝐵):𝑌 𝐿 ∧ (𝑥𝑋𝐴):𝑋𝑌) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
5021, 6, 49syl2anc 584 . . . . 5 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
5150ffnd 6638 . . . 4 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋)
5225fmpttd 7028 . . . . 5 (𝜑 → (𝑥𝑋𝐶):𝑋 𝐿)
5352ffnd 6638 . . . 4 (𝜑 → (𝑥𝑋𝐶) Fn 𝑋)
54 eqfnfv 6948 . . . 4 ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋 ∧ (𝑥𝑋𝐶) Fn 𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5551, 53, 54syl2anc 584 . . 3 (𝜑 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5648, 55mpbird 256 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))
57 cnco 22497 . . 3 (((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
584, 15, 57syl2anc 584 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
5956, 58eqeltrrd 2838 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3061   cuni 4849  cmpt 5169  ccom 5611   Fn wfn 6460  wf 6461  cfv 6465  (class class class)co 7316  Topctop 22122  TopOnctopon 22139   Cn ccn 22455
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-br 5087  df-opab 5149  df-mpt 5170  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-fv 6473  df-ov 7319  df-oprab 7320  df-mpo 7321  df-map 8666  df-top 22123  df-topon 22140  df-cn 22458
This theorem is referenced by:  cnmpt11f  22895  cnmptkp  22911  cnmptk1  22912  cnmpt1k  22913  ptunhmeo  23039  tmdgsum  23326  icchmeo  24184  evth2  24203  sinccvglem  33765  poimir  35887  broucube  35888
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