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Theorem cnmptcom 12507
 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 12471 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 409 . . . . . . . 8 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmptcom.6 . . . . . . . . . 10 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
6 cntop2 12411 . . . . . . . . . 10 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
75, 6syl 14 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
8 toptopon2 12226 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
97, 8sylib 121 . . . . . . . 8 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
10 cnf2 12414 . . . . . . . 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 1217 . . . . . . 7 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
12 eqid 2140 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1312fmpo 6107 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
14 ralcom 2597 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
1513, 14bitr3i 185 . . . . . . 7 ((𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿 ↔ ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
1611, 15sylib 121 . . . . . 6 (𝜑 → ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
17 eqid 2140 . . . . . . 7 (𝑦𝑌, 𝑥𝑋𝐴) = (𝑦𝑌, 𝑥𝑋𝐴)
1817fmpo 6107 . . . . . 6 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 ↔ (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
1916, 18sylib 121 . . . . 5 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
2019ffnd 5281 . . . 4 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) Fn (𝑌 × 𝑋))
21 fnovim 5887 . . . 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 2282 . . . . . . 7 𝑦𝑧
24 nfcv 2282 . . . . . . 7 𝑥𝑧
25 nfcv 2282 . . . . . . 7 𝑥𝑤
26 nfv 1509 . . . . . . . 8 𝑦𝜑
27 nfcv 2282 . . . . . . . . . 10 𝑦𝑥
28 nfmpo2 5847 . . . . . . . . . 10 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
2927, 28, 23nfov 5809 . . . . . . . . 9 𝑦(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧)
30 nfmpo1 5846 . . . . . . . . . 10 𝑦(𝑦𝑌, 𝑥𝑋𝐴)
3123, 30, 27nfov 5809 . . . . . . . . 9 𝑦(𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)
3229, 31nfeq 2290 . . . . . . . 8 𝑦(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)
3326, 32nfim 1552 . . . . . . 7 𝑦(𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
34 nfv 1509 . . . . . . . 8 𝑥𝜑
35 nfmpo1 5846 . . . . . . . . . 10 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
3625, 35, 24nfov 5809 . . . . . . . . 9 𝑥(𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)
37 nfmpo2 5847 . . . . . . . . . 10 𝑥(𝑦𝑌, 𝑥𝑋𝐴)
3824, 37, 25nfov 5809 . . . . . . . . 9 𝑥(𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
3936, 38nfeq 2290 . . . . . . . 8 𝑥(𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
4034, 39nfim 1552 . . . . . . 7 𝑥(𝜑 → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
41 oveq2 5790 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧))
42 oveq1 5789 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
4341, 42eqeq12d 2155 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) ↔ (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
4443imbi2d 229 . . . . . . 7 (𝑦 = 𝑧 → ((𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) ↔ (𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥))))
45 oveq1 5789 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧))
46 oveq2 5790 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
4745, 46eqeq12d 2155 . . . . . . . 8 (𝑥 = 𝑤 → ((𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥) ↔ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
4847imbi2d 229 . . . . . . 7 (𝑥 = 𝑤 → ((𝜑 → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) ↔ (𝜑 → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))))
49 rsp2 2485 . . . . . . . . 9 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 → ((𝑦𝑌𝑥𝑋) → 𝐴 𝐿))
5049, 16syl11 31 . . . . . . . 8 ((𝑦𝑌𝑥𝑋) → (𝜑𝐴 𝐿))
5112ovmpt4g 5901 . . . . . . . . . . 11 ((𝑥𝑋𝑦𝑌𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
52513com12 1186 . . . . . . . . . 10 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
5317ovmpt4g 5901 . . . . . . . . . 10 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
5452, 53eqtr4d 2176 . . . . . . . . 9 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
55543expia 1184 . . . . . . . 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 2756 . . . . . 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 1180 . . . 4 ((𝜑𝑧𝑌𝑤𝑋) → (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧) = (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
6059mpoeq3dva 5843 . . 3 (𝜑 → (𝑧𝑌, 𝑤𝑋 ↦ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑧(𝑦𝑌, 𝑥𝑋𝐴)𝑤)))
6122, 60eqtr4d 2176 . 2 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧𝑌, 𝑤𝑋 ↦ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)))
622, 1cnmpt2nd 12498 . . 3 (𝜑 → (𝑧𝑌, 𝑤𝑋𝑤) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
632, 1cnmpt1st 12497 . . 3 (𝜑 → (𝑧𝑌, 𝑤𝑋𝑧) ∈ ((𝐾 ×t 𝐽) Cn 𝐾))
642, 1, 62, 63, 5cnmpt22f 12504 . 2 (𝜑 → (𝑧𝑌, 𝑤𝑋 ↦ (𝑤(𝑥𝑋, 𝑦𝑌𝐴)𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
6561, 64eqeltrd 2217 1 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∪ cuni 3744   × cxp 4545   Fn wfn 5126  ⟶wf 5127  ‘cfv 5131  (class class class)co 5782   ∈ cmpo 5784  Topctop 12204  TopOnctopon 12217   Cn ccn 12394   ×t ctx 12461 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-topgen 12181  df-top 12205  df-topon 12218  df-bases 12250  df-cn 12397  df-tx 12462 This theorem is referenced by: (None)
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