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Theorem cnmpt2k 22290
 Description: The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypotheses
Ref Expression
cnmpt2k.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt2k.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt2k.a (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Assertion
Ref Expression
cnmpt2k (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
Distinct variable groups:   𝑥,𝑦,𝐿   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2k
Dummy variables 𝑤 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2977 . . . . 5 𝑥𝑌
2 nfcv 2977 . . . . . 6 𝑥𝑣
3 nfmpo2 7229 . . . . . 6 𝑥(𝑦𝑌, 𝑥𝑋𝐴)
4 nfcv 2977 . . . . . 6 𝑥𝑤
52, 3, 4nfov 7180 . . . . 5 𝑥(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
61, 5nfmpt 5155 . . . 4 𝑥(𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
7 nfcv 2977 . . . 4 𝑤(𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
8 nfcv 2977 . . . . . . 7 𝑦𝑣
9 nfmpo1 7228 . . . . . . 7 𝑦(𝑦𝑌, 𝑥𝑋𝐴)
10 nfcv 2977 . . . . . . 7 𝑦𝑤
118, 9, 10nfov 7180 . . . . . 6 𝑦(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
12 nfcv 2977 . . . . . 6 𝑣(𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
13 oveq1 7157 . . . . . 6 (𝑣 = 𝑦 → (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
1411, 12, 13cbvmpt 5159 . . . . 5 (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
15 oveq2 7158 . . . . . 6 (𝑤 = 𝑥 → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
1615mpteq2dv 5154 . . . . 5 (𝑤 = 𝑥 → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
1714, 16syl5eq 2868 . . . 4 (𝑤 = 𝑥 → (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
186, 7, 17cbvmpt 5159 . . 3 (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
19 simpr 487 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
20 simplr 767 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
21 cnmpt2k.k . . . . . . . . . . . 12 (𝜑𝐾 ∈ (TopOn‘𝑌))
22 cnmpt2k.j . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝑋))
23 txtopon 22193 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
2421, 22, 23syl2anc 586 . . . . . . . . . . 11 (𝜑 → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
25 cnmpt2k.a . . . . . . . . . . . . 13 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
26 cntop2 21843 . . . . . . . . . . . . 13 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑𝐿 ∈ Top)
28 toptopon2 21520 . . . . . . . . . . . 12 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2927, 28sylib 220 . . . . . . . . . . 11 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
3022, 21, 25cnmptcom 22280 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
31 cnf2 21851 . . . . . . . . . . 11 (((𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿)) → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
3224, 29, 30, 31syl3anc 1367 . . . . . . . . . 10 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
33 eqid 2821 . . . . . . . . . . 11 (𝑦𝑌, 𝑥𝑋𝐴) = (𝑦𝑌, 𝑥𝑋𝐴)
3433fmpo 7760 . . . . . . . . . 10 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 ↔ (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
3532, 34sylibr 236 . . . . . . . . 9 (𝜑 → ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
3635r19.21bi 3208 . . . . . . . 8 ((𝜑𝑦𝑌) → ∀𝑥𝑋 𝐴 𝐿)
3736r19.21bi 3208 . . . . . . 7 (((𝜑𝑦𝑌) ∧ 𝑥𝑋) → 𝐴 𝐿)
3837an32s 650 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 𝐿)
3933ovmpt4g 7291 . . . . . 6 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4019, 20, 38, 39syl3anc 1367 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4140mpteq2dva 5153 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) = (𝑦𝑌𝐴))
4241mpteq2dva 5153 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
4318, 42syl5eq 2868 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
44 eqid 2821 . . . . 5 (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩))
4544xkoinjcn 22289 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ↑ko 𝐾)))
4622, 21, 45syl2anc 586 . . 3 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ↑ko 𝐾)))
4732feqmptd 6727 . . . 4 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)))
4847, 30eqeltrrd 2914 . . 3 (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
49 fveq2 6664 . . . 4 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩))
50 df-ov 7153 . . . 4 (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩)
5149, 50syl6eqr 2874 . . 3 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
5222, 21, 24, 46, 48, 51cnmptk1 22283 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) ∈ (𝐽 Cn (𝐿ko 𝐾)))
5343, 52eqeltrrd 2914 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ⟨cop 4566  ∪ cuni 4831   ↦ cmpt 5138   × cxp 5547  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  Topctop 21495  TopOnctopon 21512   Cn ccn 21826   ×t ctx 22162   ↑ko cxko 22163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-fin 8507  df-fi 8869  df-rest 16690  df-topgen 16711  df-top 21496  df-topon 21513  df-bases 21548  df-cn 21829  df-cnp 21830  df-cmp 21989  df-tx 22164  df-xko 22165 This theorem is referenced by:  xkocnv  22416  xkohmeo  22417
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