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Theorem cnmpt11 13450
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 110 . . . . . . . . 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 13372 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋𝑌)
62, 3, 4, 5syl3anc 1238 . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
7 eqid 2177 . . . . . . . . . . . 12 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
87fmpt 5662 . . . . . . . . . . 11 (∀𝑥𝑋 𝐴𝑌 ↔ (𝑥𝑋𝐴):𝑋𝑌)
96, 8sylibr 134 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
109r19.21bi 2565 . . . . . . . . 9 ((𝜑𝑥𝑋) → 𝐴𝑌)
117fvmpt2 5595 . . . . . . . . 9 ((𝑥𝑋𝐴𝑌) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
121, 10, 11syl2anc 411 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
1312fveq2d 5515 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = ((𝑦𝑌𝐵)‘𝐴))
14 eqid 2177 . . . . . . . 8 (𝑦𝑌𝐵) = (𝑦𝑌𝐵)
15 cnmpt11.c . . . . . . . 8 (𝑦 = 𝐴𝐵 = 𝐶)
1615eleq1d 2246 . . . . . . . . 9 (𝑦 = 𝐴 → (𝐵 𝐿𝐶 𝐿))
17 cnmpt11.b . . . . . . . . . . . . . 14 (𝜑 → (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿))
18 cntop2 13369 . . . . . . . . . . . . . 14 ((𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿) → 𝐿 ∈ Top)
1917, 18syl 14 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ Top)
20 eqid 2177 . . . . . . . . . . . . . 14 𝐿 = 𝐿
2120toptopon 13183 . . . . . . . . . . . . 13 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2219, 21sylib 122 . . . . . . . . . . . 12 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
23 cnf2 13372 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐵):𝑌 𝐿)
243, 22, 17, 23syl3anc 1238 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌𝐵):𝑌 𝐿)
2514fmpt 5662 . . . . . . . . . . 11 (∀𝑦𝑌 𝐵 𝐿 ↔ (𝑦𝑌𝐵):𝑌 𝐿)
2624, 25sylibr 134 . . . . . . . . . 10 (𝜑 → ∀𝑦𝑌 𝐵 𝐿)
2726adantr 276 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 𝐿)
2816, 27, 10rspcdva 2846 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝐶 𝐿)
2914, 15, 10, 28fvmptd3 5605 . . . . . . 7 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘𝐴) = 𝐶)
3013, 29eqtrd 2210 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)) = 𝐶)
31 fvco3 5583 . . . . . . 7 (((𝑥𝑋𝐴):𝑋𝑌𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
326, 31sylan 283 . . . . . 6 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑦𝑌𝐵)‘((𝑥𝑋𝐴)‘𝑥)))
33 eqid 2177 . . . . . . . 8 (𝑥𝑋𝐶) = (𝑥𝑋𝐶)
3433fvmpt2 5595 . . . . . . 7 ((𝑥𝑋𝐶 𝐿) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
351, 28, 34syl2anc 411 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐶)‘𝑥) = 𝐶)
3630, 32, 353eqtr4d 2220 . . . . 5 ((𝜑𝑥𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
3736ralrimiva 2550 . . . 4 (𝜑 → ∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥))
38 nfv 1528 . . . . 5 𝑧(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥)
39 nfcv 2319 . . . . . . . 8 𝑥(𝑦𝑌𝐵)
40 nfmpt1 4093 . . . . . . . 8 𝑥(𝑥𝑋𝐴)
4139, 40nfco 4788 . . . . . . 7 𝑥((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))
42 nfcv 2319 . . . . . . 7 𝑥𝑧
4341, 42nffv 5521 . . . . . 6 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧)
44 nfmpt1 4093 . . . . . . 7 𝑥(𝑥𝑋𝐶)
4544, 42nffv 5521 . . . . . 6 𝑥((𝑥𝑋𝐶)‘𝑧)
4643, 45nfeq 2327 . . . . 5 𝑥(((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)
47 fveq2 5511 . . . . . 6 (𝑥 = 𝑧 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧))
48 fveq2 5511 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝑋𝐶)‘𝑥) = ((𝑥𝑋𝐶)‘𝑧))
4947, 48eqeq12d 2192 . . . . 5 (𝑥 = 𝑧 → ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5038, 46, 49cbvral 2699 . . . 4 (∀𝑥𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑥) = ((𝑥𝑋𝐶)‘𝑥) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
5137, 50sylib 122 . . 3 (𝜑 → ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧))
52 fco 5377 . . . . . 6 (((𝑦𝑌𝐵):𝑌 𝐿 ∧ (𝑥𝑋𝐴):𝑋𝑌) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
5324, 6, 52syl2anc 411 . . . . 5 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)):𝑋 𝐿)
5453ffnd 5362 . . . 4 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋)
5528fmpttd 5667 . . . . 5 (𝜑 → (𝑥𝑋𝐶):𝑋 𝐿)
5655ffnd 5362 . . . 4 (𝜑 → (𝑥𝑋𝐶) Fn 𝑋)
57 eqfnfv 5609 . . . 4 ((((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) Fn 𝑋 ∧ (𝑥𝑋𝐶) Fn 𝑋) → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5854, 56, 57syl2anc 411 . . 3 (𝜑 → (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶) ↔ ∀𝑧𝑋 (((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴))‘𝑧) = ((𝑥𝑋𝐶)‘𝑧)))
5951, 58mpbird 167 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))
60 cnco 13388 . . 3 (((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿)) → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
614, 17, 60syl2anc 411 . 2 (𝜑 → ((𝑦𝑌𝐵) ∘ (𝑥𝑋𝐴)) ∈ (𝐽 Cn 𝐿))
6259, 61eqeltrrd 2255 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455   cuni 3807  cmpt 4061  ccom 4627   Fn wfn 5207  wf 5208  cfv 5212  (class class class)co 5869  Topctop 13162  TopOnctopon 13175   Cn ccn 13352
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-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
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-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-top 13163  df-topon 13176  df-cn 13355
This theorem is referenced by:  cnmpt11f  13451
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