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Theorem cnmpt2k 21693
 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 (𝐿 ^ko 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 2902 . . . . 5 𝑥𝑌
2 nfcv 2902 . . . . . 6 𝑥𝑣
3 nfmpt22 6888 . . . . . 6 𝑥(𝑦𝑌, 𝑥𝑋𝐴)
4 nfcv 2902 . . . . . 6 𝑥𝑤
52, 3, 4nfov 6839 . . . . 5 𝑥(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
61, 5nfmpt 4898 . . . 4 𝑥(𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
7 nfcv 2902 . . . 4 𝑤(𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
8 nfcv 2902 . . . . . . 7 𝑦𝑣
9 nfmpt21 6887 . . . . . . 7 𝑦(𝑦𝑌, 𝑥𝑋𝐴)
10 nfcv 2902 . . . . . . 7 𝑦𝑤
118, 9, 10nfov 6839 . . . . . 6 𝑦(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
12 nfcv 2902 . . . . . 6 𝑣(𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
13 oveq1 6820 . . . . . 6 (𝑣 = 𝑦 → (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
1411, 12, 13cbvmpt 4901 . . . . 5 (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
15 oveq2 6821 . . . . . 6 (𝑤 = 𝑥 → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
1615mpteq2dv 4897 . . . . 5 (𝑤 = 𝑥 → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
1714, 16syl5eq 2806 . . . 4 (𝑤 = 𝑥 → (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
186, 7, 17cbvmpt 4901 . . 3 (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
19 simpr 479 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
20 simplr 809 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
21 cnmpt2k.k . . . . . . . . . . . 12 (𝜑𝐾 ∈ (TopOn‘𝑌))
22 cnmpt2k.j . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝑋))
23 txtopon 21596 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
2421, 22, 23syl2anc 696 . . . . . . . . . . 11 (𝜑 → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
25 cnmpt2k.a . . . . . . . . . . . . 13 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
26 cntop2 21247 . . . . . . . . . . . . 13 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑𝐿 ∈ Top)
28 eqid 2760 . . . . . . . . . . . . 13 𝐿 = 𝐿
2928toptopon 20924 . . . . . . . . . . . 12 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
3027, 29sylib 208 . . . . . . . . . . 11 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
3122, 21, 25cnmptcom 21683 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
32 cnf2 21255 . . . . . . . . . . 11 (((𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿)) → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
3324, 30, 31, 32syl3anc 1477 . . . . . . . . . 10 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
34 eqid 2760 . . . . . . . . . . 11 (𝑦𝑌, 𝑥𝑋𝐴) = (𝑦𝑌, 𝑥𝑋𝐴)
3534fmpt2 7405 . . . . . . . . . 10 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 ↔ (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
3633, 35sylibr 224 . . . . . . . . 9 (𝜑 → ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
3736r19.21bi 3070 . . . . . . . 8 ((𝜑𝑦𝑌) → ∀𝑥𝑋 𝐴 𝐿)
3837r19.21bi 3070 . . . . . . 7 (((𝜑𝑦𝑌) ∧ 𝑥𝑋) → 𝐴 𝐿)
3938an32s 881 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 𝐿)
4034ovmpt4g 6948 . . . . . 6 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4119, 20, 39, 40syl3anc 1477 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4241mpteq2dva 4896 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) = (𝑦𝑌𝐴))
4342mpteq2dva 4896 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
4418, 43syl5eq 2806 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
45 eqid 2760 . . . . 5 (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩))
4645xkoinjcn 21692 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ^ko 𝐾)))
4722, 21, 46syl2anc 696 . . 3 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ^ko 𝐾)))
4833feqmptd 6411 . . . 4 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)))
4948, 31eqeltrrd 2840 . . 3 (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
50 fveq2 6352 . . . 4 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩))
51 df-ov 6816 . . . 4 (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩)
5250, 51syl6eqr 2812 . . 3 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
5322, 21, 24, 47, 49, 52cnmptk1 21686 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
5444, 53eqeltrrd 2840 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ⟨cop 4327  ∪ cuni 4588   ↦ cmpt 4881   × cxp 5264  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815  Topctop 20900  TopOnctopon 20917   Cn ccn 21230   ×t ctx 21565   ^ko cxko 21566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-fin 8125  df-fi 8482  df-rest 16285  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cn 21233  df-cnp 21234  df-cmp 21392  df-tx 21567  df-xko 21568 This theorem is referenced by:  xkocnv  21819  xkohmeo  21820
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