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Theorem cnmptcom 21702
Description: The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
Hypotheses
Ref Expression
cnmptcom.3 (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptcom.4 (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptcom.6 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Assertion
Ref Expression
cnmptcom (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
Distinct variable groups:   𝑥,𝑦,𝐿   𝑥,𝑋,𝑦   𝜑,𝑥,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmptcom
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnmptcom.3 . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmptcom.4 . . . . . . . . 9 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 txtopon 21615 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 573 . . . . . . . 8 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmptcom.6 . . . . . . . . . 10 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
6 cntop2 21266 . . . . . . . . . 10 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
75, 6syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
8 eqid 2771 . . . . . . . . . 10 𝐿 = 𝐿
98toptopon 20942 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
107, 9sylib 208 . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
11 cnf2 21274 . . . . . . . 8 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
124, 10, 5, 11syl3anc 1476 . . . . . . 7 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
13 eqid 2771 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1413fmpt2 7387 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
15 ralcom 3246 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
1614, 15bitr3i 266 . . . . . . 7 ((𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿 ↔ ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
1712, 16sylib 208 . . . . . 6 (𝜑 → ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
18 eqid 2771 . . . . . . 7 (𝑦𝑌, 𝑥𝑋𝐴) = (𝑦𝑌, 𝑥𝑋𝐴)
1918fmpt2 7387 . . . . . 6 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 ↔ (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
2017, 19sylib 208 . . . . 5 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
21 ffn 6185 . . . . 5 ((𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿 → (𝑦𝑌, 𝑥𝑋𝐴) Fn (𝑌 × 𝑋))
2220, 21syl 17 . . . 4 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) Fn (𝑌 × 𝑋))
23 fnov 6915 . . . 4 ((𝑦𝑌, 𝑥𝑋𝐴) Fn (𝑌 × 𝑋) ↔ (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
2422, 23sylib 208 . . 3 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
25 nfcv 2913 . . . . . . 7 𝑦𝑧
26 nfcv 2913 . . . . . . 7 𝑥𝑧
27 nfcv 2913 . . . . . . 7 𝑥𝑤
28 nfv 1995 . . . . . . . 8 𝑦𝜑
29 nfcv 2913 . . . . . . . . . 10 𝑦𝑥
30 nfmpt22 6870 . . . . . . . . . 10 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
3129, 30, 25nfov 6821 . . . . . . . . 9 𝑦(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧)
32 nfmpt21 6869 . . . . . . . . . 10 𝑦(𝑦𝑌, 𝑥𝑋𝐴)
3325, 32, 29nfov 6821 . . . . . . . . 9 𝑦(𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)
3431, 33nfeq 2925 . . . . . . . 8 𝑦(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)
3528, 34nfim 1977 . . . . . . 7 𝑦(𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
36 nfv 1995 . . . . . . . 8 𝑥𝜑
37 nfmpt21 6869 . . . . . . . . . 10 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
3827, 37, 26nfov 6821 . . . . . . . . 9 𝑥(𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)
39 nfmpt22 6870 . . . . . . . . . 10 𝑥(𝑦𝑌, 𝑥𝑋𝐴)
4026, 39, 27nfov 6821 . . . . . . . . 9 𝑥(𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
4138, 40nfeq 2925 . . . . . . . 8 𝑥(𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
4236, 41nfim 1977 . . . . . . 7 𝑥(𝜑 → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
43 oveq2 6801 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧))
44 oveq1 6800 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
4543, 44eqeq12d 2786 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) ↔ (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
4645imbi2d 329 . . . . . . 7 (𝑦 = 𝑧 → ((𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) ↔ (𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥))))
47 oveq1 6800 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧))
48 oveq2 6801 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
4947, 48eqeq12d 2786 . . . . . . . 8 (𝑥 = 𝑤 → ((𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥) ↔ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
5049imbi2d 329 . . . . . . 7 (𝑥 = 𝑤 → ((𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) ↔ (𝜑 → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))))
51 rsp2 3085 . . . . . . . . 9 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 → ((𝑦𝑌𝑥𝑋) → 𝐴 𝐿))
5251, 17syl11 33 . . . . . . . 8 ((𝑦𝑌𝑥𝑋) → (𝜑𝐴 𝐿))
5313ovmpt4g 6930 . . . . . . . . . . 11 ((𝑥𝑋𝑦𝑌𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
54533com12 1117 . . . . . . . . . 10 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
5518ovmpt4g 6930 . . . . . . . . . 10 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
5654, 55eqtr4d 2808 . . . . . . . . 9 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
57563expia 1114 . . . . . . . 8 ((𝑦𝑌𝑥𝑋) → (𝐴 𝐿 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
5852, 57syld 47 . . . . . . 7 ((𝑦𝑌𝑥𝑋) → (𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
5925, 26, 27, 35, 42, 46, 50, 58vtocl2gaf 3424 . . . . . 6 ((𝑧𝑌𝑤𝑋) → (𝜑 → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
6059com12 32 . . . . 5 (𝜑 → ((𝑧𝑌𝑤𝑋) → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
61603impib 1108 . . . 4 ((𝜑𝑧𝑌𝑤𝑋) → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
6261mpt2eq3dva 6866 . . 3 (𝜑 → (𝑧𝑌, 𝑤𝑋 ↦ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
6324, 62eqtr4d 2808 . 2 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)))
642, 1cnmpt2nd 21693 . . 3 (𝜑 → (𝑧𝑌, 𝑤𝑋𝑤) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
652, 1cnmpt1st 21692 . . 3 (𝜑 → (𝑧𝑌, 𝑤𝑋𝑧) ∈ ((𝐾 ×t 𝐽) Cn 𝐾))
662, 1, 64, 65, 5cnmpt22f 21699 . 2 (𝜑 → (𝑧𝑌, 𝑤𝑋 ↦ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
6763, 66eqeltrd 2850 1 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061   cuni 4574   × cxp 5247   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  Topctop 20918  TopOnctopon 20935   Cn ccn 21249   ×t ctx 21584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-cn 21252  df-tx 21586
This theorem is referenced by:  cnmpt2k  21712  htpycc  22999
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