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Theorem cnmpt2k 23605
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 2899 . . . . 5 β„²π‘₯π‘Œ
2 nfcv 2899 . . . . . 6 β„²π‘₯𝑣
3 nfmpo2 7501 . . . . . 6 β„²π‘₯(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
4 nfcv 2899 . . . . . 6 β„²π‘₯𝑀
52, 3, 4nfov 7450 . . . . 5 β„²π‘₯(𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
61, 5nfmpt 5255 . . . 4 β„²π‘₯(𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
7 nfcv 2899 . . . 4 Ⅎ𝑀(𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
8 nfcv 2899 . . . . . . 7 Ⅎ𝑦𝑣
9 nfmpo1 7500 . . . . . . 7 Ⅎ𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
10 nfcv 2899 . . . . . . 7 Ⅎ𝑦𝑀
118, 9, 10nfov 7450 . . . . . 6 Ⅎ𝑦(𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
12 nfcv 2899 . . . . . 6 Ⅎ𝑣(𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
13 oveq1 7427 . . . . . 6 (𝑣 = 𝑦 β†’ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
1411, 12, 13cbvmpt 5259 . . . . 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 7428 . . . . . 6 (𝑀 = π‘₯ β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
1615mpteq2dv 5250 . . . . 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 2780 . . . 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 5259 . . 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 768 . . . . . 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 23508 . . . . . . . . . . . 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 23158 . . . . . . . . . . . . 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 22833 . . . . . . . . . . . 12 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
2927, 28sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
3022, 21, 25cnmptcom 23595 . . . . . . . . . . 11 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿))
31 cnf2 23166 . . . . . . . . . . 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 1369 . . . . . . . . . 10 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴):(π‘Œ Γ— 𝑋)⟢βˆͺ 𝐿)
33 eqid 2728 . . . . . . . . . . 11 (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
3433fmpo 8072 . . . . . . . . . 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 3245 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ π‘Œ) β†’ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿)
3736r19.21bi 3245 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ βˆͺ 𝐿)
3837an32s 651 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ 𝐴 ∈ βˆͺ 𝐿)
3933ovmpt4g 7568 . . . . . 6 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = 𝐴)
4019, 20, 38, 39syl3anc 1369 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = 𝐴)
4140mpteq2dva 5248 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)) = (𝑦 ∈ π‘Œ ↦ 𝐴))
4241mpteq2dva 5248 . . 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 2780 . 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 2728 . . . . 5 (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ βŸ¨π‘£, π‘€βŸ©)) = (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ βŸ¨π‘£, π‘€βŸ©))
4544xkoinjcn 23604 . . . 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 6967 . . . 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 2830 . . 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 6897 . . . 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 7423 . . . 4 (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀) = ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜βŸ¨π‘£, π‘€βŸ©)
5149, 50eqtr4di 2786 . . 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 23598 . 2 (πœ‘ β†’ (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
5343, 52eqeltrrd 2830 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 1534   ∈ wcel 2099  βˆ€wral 3058  βŸ¨cop 4635  βˆͺ cuni 4908   ↦ cmpt 5231   Γ— cxp 5676  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420   ∈ cmpo 7422  Topctop 22808  TopOnctopon 22825   Cn ccn 23141   Γ—t ctx 23477   ↑ko cxko 23478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-fin 8968  df-fi 9435  df-rest 17404  df-topgen 17425  df-top 22809  df-topon 22826  df-bases 22862  df-cn 23144  df-cnp 23145  df-cmp 23304  df-tx 23479  df-xko 23480
This theorem is referenced by:  xkocnv  23731  xkohmeo  23732
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