MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt11 Structured version   Visualization version   GIF version

Theorem cnmpt11 21668
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 479 . . . . . . . . 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 21255 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
62, 3, 4, 5syl3anc 1477 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
7 eqid 2760 . . . . . . . . . . . 12 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
87fmpt 6544 . . . . . . . . . . 11 (∀𝑥𝑋 𝐴𝑌 ↔ (𝑥𝑋𝐴):𝑋𝑌)
96, 8sylibr 224 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
109r19.21bi 3070 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐴𝑌)
117fvmpt2 6453 . . . . . . . . 9 ((𝑥𝑋𝐴𝑌) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
121, 10, 11syl2anc 696 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
1312fveq2d 6356 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = ((𝑦𝑌𝐵)‘𝐴))
14 cnmpt11.c . . . . . . . . . 10 (𝑦 = 𝐴𝐵 = 𝐶)
1514eleq1d 2824 . . . . . . . . 9 (𝑦 = 𝐴 → (𝐵 𝐿𝐶 𝐿))
16 cnmpt11.b . . . . . . . . . . . . . 14 (𝜑 → (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿))
17 cntop2 21247 . . . . . . . . . . . . . 14 ((𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
1816, 17syl 17 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ Top)
19 eqid 2760 . . . . . . . . . . . . . 14 𝐿 = 𝐿
2019toptopon 20924 . . . . . . . . . . . . 13 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2118, 20sylib 208 . . . . . . . . . . . 12 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
22 cnf2 21255 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐵):𝑌 𝐿)
233, 21, 16, 22syl3anc 1477 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌𝐵):𝑌 𝐿)
24 eqid 2760 . . . . . . . . . . . 12 (𝑦𝑌𝐵) = (𝑦𝑌𝐵)
2524fmpt 6544 . . . . . . . . . . 11 (∀𝑦𝑌 𝐵 𝐿 ↔ (𝑦𝑌𝐵):𝑌 𝐿)
2623, 25sylibr 224 . . . . . . . . . 10 (𝜑 → ∀𝑦𝑌 𝐵 𝐿)
2726adantr 472 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 𝐿)
2815, 27, 10rspcdva 3455 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝐶 𝐿)
2914, 24fvmptg 6442 . . . . . . . 8 ((𝐴𝑌𝐶 𝐿) → ((𝑦𝑌𝐵)‘𝐴) = 𝐶)
3010, 28, 29syl2anc 696 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘𝐴) = 𝐶)
3113, 30eqtrd 2794 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = 𝐶)
32 fvco3 6437 . . . . . . 7 (((𝑥𝑋𝐴):𝑋𝑌𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
336, 32sylan 489 . . . . . 6 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
34 eqid 2760 . . . . . . . 8 (𝑥𝑋𝐶) = (𝑥𝑋𝐶)
3534fvmpt2 6453 . . . . . . 7 ((𝑥𝑋𝐶 𝐿) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
361, 28, 35syl2anc 696 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
3731, 33, 363eqtr4d 2804 . . . . 5 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
3837ralrimiva 3104 . . . 4 (𝜑 → ∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
39 nfv 1992 . . . . 5 𝑧(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥)
40 nfcv 2902 . . . . . . . 8 𝑥(𝑦𝑌𝐵)
41 nfmpt1 4899 . . . . . . . 8 𝑥(𝑥𝑋𝐴)
4240, 41nfco 5443 . . . . . . 7 𝑥((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))
43 nfcv 2902 . . . . . . 7 𝑥𝑧
4442, 43nffv 6359 . . . . . 6 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧)
45 nfmpt1 4899 . . . . . . 7 𝑥(𝑥𝑋𝐶)
4645, 43nffv 6359 . . . . . 6 𝑥((𝑥𝑋𝐶)‘𝑧)
4744, 46nfeq 2914 . . . . 5 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)
48 fveq2 6352 . . . . . 6 (𝑥 = 𝑧 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧))
49 fveq2 6352 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝑋𝐶)‘𝑥) = ((𝑥𝑋𝐶)‘𝑧))
5048, 49eqeq12d 2775 . . . . 5 (𝑥 = 𝑧 → ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5139, 47, 50cbvral 3306 . . . 4 (∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
5238, 51sylib 208 . . 3 (𝜑 → ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
53 fco 6219 . . . . . 6 (((𝑦𝑌𝐵):𝑌 𝐿 ∧ (𝑥𝑋𝐴):𝑋𝑌) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
5423, 6, 53syl2anc 696 . . . . 5 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
55 ffn 6206 . . . . 5 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋)
5654, 55syl 17 . . . 4 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋)
5728, 34fmptd 6548 . . . . 5 (𝜑 → (𝑥𝑋𝐶):𝑋 𝐿)
58 ffn 6206 . . . . 5 ((𝑥𝑋𝐶):𝑋 𝐿 → (𝑥𝑋𝐶) Fn 𝑋)
5957, 58syl 17 . . . 4 (𝜑 → (𝑥𝑋𝐶) Fn 𝑋)
60 eqfnfv 6474 . . . 4 ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋 ∧ (𝑥𝑋𝐶) Fn 𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
6156, 59, 60syl2anc 696 . . 3 (𝜑 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
6252, 61mpbird 247 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))
63 cnco 21272 . . 3 (((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
644, 16, 63syl2anc 696 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
6562, 64eqeltrrd 2840 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050   cuni 4588  cmpt 4881  ccom 5270   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  Topctop 20900  TopOnctopon 20917   Cn ccn 21230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-top 20901  df-topon 20918  df-cn 21233
This theorem is referenced by:  cnmpt11f  21669  cnmptkp  21685  cnmptk1  21686  cnmpt1k  21687  ptunhmeo  21813  tmdgsum  22100  icchmeo  22941  evth2  22960  sinccvglem  31873  poimir  33755  broucube  33756
  Copyright terms: Public domain W3C validator