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Theorem cnmptcom 13378
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 13342 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 411 . . . . . . . 8 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmptcom.6 . . . . . . . . . 10 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
6 cntop2 13282 . . . . . . . . . 10 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
75, 6syl 14 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
8 toptopon2 13097 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
97, 8sylib 122 . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
10 cnf2 13285 . . . . . . . 8 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
114, 9, 5, 10syl3anc 1238 . . . . . . 7 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
12 eqid 2175 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1312fmpo 6192 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
14 ralcom 2638 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
1513, 14bitr3i 186 . . . . . . 7 ((𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿 ↔ ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
1611, 15sylib 122 . . . . . 6 (𝜑 → ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
17 eqid 2175 . . . . . . 7 (𝑦𝑌, 𝑥𝑋𝐴) = (𝑦𝑌, 𝑥𝑋𝐴)
1817fmpo 6192 . . . . . 6 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 ↔ (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
1916, 18sylib 122 . . . . 5 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
2019ffnd 5358 . . . 4 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) Fn (𝑌 × 𝑋))
21 fnovim 5973 . . . 4 ((𝑦𝑌, 𝑥𝑋𝐴) Fn (𝑌 × 𝑋) → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
2220, 21syl 14 . . 3 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
23 nfcv 2317 . . . . . . 7 𝑦𝑧
24 nfcv 2317 . . . . . . 7 𝑥𝑧
25 nfcv 2317 . . . . . . 7 𝑥𝑤
26 nfv 1526 . . . . . . . 8 𝑦𝜑
27 nfcv 2317 . . . . . . . . . 10 𝑦𝑥
28 nfmpo2 5933 . . . . . . . . . 10 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
2927, 28, 23nfov 5895 . . . . . . . . 9 𝑦(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧)
30 nfmpo1 5932 . . . . . . . . . 10 𝑦(𝑦𝑌, 𝑥𝑋𝐴)
3123, 30, 27nfov 5895 . . . . . . . . 9 𝑦(𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)
3229, 31nfeq 2325 . . . . . . . 8 𝑦(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)
3326, 32nfim 1570 . . . . . . 7 𝑦(𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
34 nfv 1526 . . . . . . . 8 𝑥𝜑
35 nfmpo1 5932 . . . . . . . . . 10 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
3625, 35, 24nfov 5895 . . . . . . . . 9 𝑥(𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)
37 nfmpo2 5933 . . . . . . . . . 10 𝑥(𝑦𝑌, 𝑥𝑋𝐴)
3824, 37, 25nfov 5895 . . . . . . . . 9 𝑥(𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
3936, 38nfeq 2325 . . . . . . . 8 𝑥(𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
4034, 39nfim 1570 . . . . . . 7 𝑥(𝜑 → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
41 oveq2 5873 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧))
42 oveq1 5872 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
4341, 42eqeq12d 2190 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) ↔ (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
4443imbi2d 230 . . . . . . 7 (𝑦 = 𝑧 → ((𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) ↔ (𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥))))
45 oveq1 5872 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧))
46 oveq2 5873 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
4745, 46eqeq12d 2190 . . . . . . . 8 (𝑥 = 𝑤 → ((𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥) ↔ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
4847imbi2d 230 . . . . . . 7 (𝑥 = 𝑤 → ((𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) ↔ (𝜑 → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))))
49 rsp2 2525 . . . . . . . . 9 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 → ((𝑦𝑌𝑥𝑋) → 𝐴 𝐿))
5049, 16syl11 31 . . . . . . . 8 ((𝑦𝑌𝑥𝑋) → (𝜑𝐴 𝐿))
5112ovmpt4g 5987 . . . . . . . . . . 11 ((𝑥𝑋𝑦𝑌𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
52513com12 1207 . . . . . . . . . 10 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
5317ovmpt4g 5987 . . . . . . . . . 10 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
5452, 53eqtr4d 2211 . . . . . . . . 9 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
55543expia 1205 . . . . . . . 8 ((𝑦𝑌𝑥𝑋) → (𝐴 𝐿 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
5650, 55syld 45 . . . . . . 7 ((𝑦𝑌𝑥𝑋) → (𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
5723, 24, 25, 33, 40, 44, 48, 56vtocl2gaf 2802 . . . . . 6 ((𝑧𝑌𝑤𝑋) → (𝜑 → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
5857com12 30 . . . . 5 (𝜑 → ((𝑧𝑌𝑤𝑋) → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
59583impib 1201 . . . 4 ((𝜑𝑧𝑌𝑤𝑋) → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
6059mpoeq3dva 5929 . . 3 (𝜑 → (𝑧𝑌, 𝑤𝑋 ↦ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
6122, 60eqtr4d 2211 . 2 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)))
622, 1cnmpt2nd 13369 . . 3 (𝜑 → (𝑧𝑌, 𝑤𝑋𝑤) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
632, 1cnmpt1st 13368 . . 3 (𝜑 → (𝑧𝑌, 𝑤𝑋𝑧) ∈ ((𝐾 ×t 𝐽) Cn 𝐾))
642, 1, 62, 63, 5cnmpt22f 13375 . 2 (𝜑 → (𝑧𝑌, 𝑤𝑋 ↦ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
6561, 64eqeltrd 2252 1 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2146  wral 2453   cuni 3805   × cxp 4618   Fn wfn 5203  wf 5204  cfv 5208  (class class class)co 5865  cmpo 5867  Topctop 13075  TopOnctopon 13088   Cn ccn 13265   ×t ctx 13332
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-topgen 12640  df-top 13076  df-topon 13089  df-bases 13121  df-cn 13268  df-tx 13333
This theorem is referenced by: (None)
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