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Theorem cnmpt2k 23543
Description: The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypotheses
Ref Expression
cnmpt2k.j (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmpt2k.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmpt2k.a (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
Assertion
Ref Expression
cnmpt2k (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
Distinct variable groups:   π‘₯,𝑦,𝐿   πœ‘,π‘₯,𝑦   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐽(π‘₯,𝑦)   𝐾(π‘₯,𝑦)
Proof of Theorem cnmpt2k
Dummy variables 𝑀 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2897 . . . . 5 β„²π‘₯π‘Œ
2 nfcv 2897 . . . . . 6 β„²π‘₯𝑣
3 nfmpo2 7485 . . . . . 6 β„²π‘₯(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
4 nfcv 2897 . . . . . 6 β„²π‘₯𝑀
52, 3, 4nfov 7434 . . . . 5 β„²π‘₯(𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
61, 5nfmpt 5248 . . . 4 β„²π‘₯(𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
7 nfcv 2897 . . . 4 Ⅎ𝑀(𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
8 nfcv 2897 . . . . . . 7 Ⅎ𝑦𝑣
9 nfmpo1 7484 . . . . . . 7 Ⅎ𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
10 nfcv 2897 . . . . . . 7 Ⅎ𝑦𝑀
118, 9, 10nfov 7434 . . . . . 6 Ⅎ𝑦(𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
12 nfcv 2897 . . . . . 6 Ⅎ𝑣(𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
13 oveq1 7411 . . . . . 6 (𝑣 = 𝑦 β†’ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
1411, 12, 13cbvmpt 5252 . . . . 5 (𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)) = (𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
15 oveq2 7412 . . . . . 6 (𝑀 = π‘₯ β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
1615mpteq2dv 5243 . . . . 5 (𝑀 = π‘₯ β†’ (𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)) = (𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)))
1714, 16eqtrid 2778 . . . 4 (𝑀 = π‘₯ β†’ (𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)) = (𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)))
186, 7, 17cbvmpt 5252 . . 3 (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))) = (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)))
19 simpr 484 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ π‘Œ)
20 simplr 766 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ π‘₯ ∈ 𝑋)
21 cnmpt2k.k . . . . . . . . . . . 12 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
22 cnmpt2k.j . . . . . . . . . . . 12 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
23 txtopon 23446 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐾 Γ—t 𝐽) ∈ (TopOnβ€˜(π‘Œ Γ— 𝑋)))
2421, 22, 23syl2anc 583 . . . . . . . . . . 11 (πœ‘ β†’ (𝐾 Γ—t 𝐽) ∈ (TopOnβ€˜(π‘Œ Γ— 𝑋)))
25 cnmpt2k.a . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
26 cntop2 23096 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿) β†’ 𝐿 ∈ Top)
2725, 26syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐿 ∈ Top)
28 toptopon2 22771 . . . . . . . . . . . 12 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
2927, 28sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
3022, 21, 25cnmptcom 23533 . . . . . . . . . . 11 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿))
31 cnf2 23104 . . . . . . . . . . 11 (((𝐾 Γ—t 𝐽) ∈ (TopOnβ€˜(π‘Œ Γ— 𝑋)) ∧ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿) ∧ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿)) β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴):(π‘Œ Γ— 𝑋)⟢βˆͺ 𝐿)
3224, 29, 30, 31syl3anc 1368 . . . . . . . . . 10 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴):(π‘Œ Γ— 𝑋)⟢βˆͺ 𝐿)
33 eqid 2726 . . . . . . . . . . 11 (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
3433fmpo 8050 . . . . . . . . . 10 (βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿 ↔ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴):(π‘Œ Γ— 𝑋)⟢βˆͺ 𝐿)
3532, 34sylibr 233 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿)
3635r19.21bi 3242 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ π‘Œ) β†’ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿)
3736r19.21bi 3242 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ βˆͺ 𝐿)
3837an32s 649 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐿)
3933ovmpt4g 7550 . . . . . 6 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = 𝐴)
4019, 20, 38, 39syl3anc 1368 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = 𝐴)
4140mpteq2dva 5241 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)) = (𝑦 ∈ π‘Œ ↦ 𝐴))
4241mpteq2dva 5241 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))) = (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)))
4318, 42eqtrid 2778 . 2 (πœ‘ β†’ (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))) = (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)))
44 eqid 2726 . . . . 5 (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ βŸ¨π‘£, π‘€βŸ©)) = (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ βŸ¨π‘£, π‘€βŸ©))
4544xkoinjcn 23542 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ βŸ¨π‘£, π‘€βŸ©)) ∈ (𝐽 Cn ((𝐾 Γ—t 𝐽) ↑ko 𝐾)))
4622, 21, 45syl2anc 583 . . 3 (πœ‘ β†’ (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ βŸ¨π‘£, π‘€βŸ©)) ∈ (𝐽 Cn ((𝐾 Γ—t 𝐽) ↑ko 𝐾)))
4732feqmptd 6953 . . . 4 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ (π‘Œ Γ— 𝑋) ↦ ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘§)))
4847, 30eqeltrrd 2828 . . 3 (πœ‘ β†’ (𝑧 ∈ (π‘Œ Γ— 𝑋) ↦ ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘§)) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿))
49 fveq2 6884 . . . 4 (𝑧 = βŸ¨π‘£, π‘€βŸ© β†’ ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘§) = ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜βŸ¨π‘£, π‘€βŸ©))
50 df-ov 7407 . . . 4 (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀) = ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜βŸ¨π‘£, π‘€βŸ©)
5149, 50eqtr4di 2784 . . 3 (𝑧 = βŸ¨π‘£, π‘€βŸ© β†’ ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘§) = (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
5222, 21, 24, 46, 48, 51cnmptk1 23536 . 2 (πœ‘ β†’ (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
5343, 52eqeltrrd 2828 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝑦 ∈ π‘Œ ↦ 𝐴)) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βŸ¨cop 4629  βˆͺ cuni 4902   ↦ cmpt 5224   Γ— cxp 5667  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  Topctop 22746  TopOnctopon 22763   Cn ccn 23079   Γ—t ctx 23415   ↑ko cxko 23416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8464  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-fin 8942  df-fi 9405  df-rest 17375  df-topgen 17396  df-top 22747  df-topon 22764  df-bases 22800  df-cn 23082  df-cnp 23083  df-cmp 23242  df-tx 23417  df-xko 23418
This theorem is referenced by:  xkocnv  23669  xkohmeo  23670
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