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Theorem cnmpt2k 23062
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 2904 . . . . 5 β„²π‘₯π‘Œ
2 nfcv 2904 . . . . . 6 β„²π‘₯𝑣
3 nfmpo2 7442 . . . . . 6 β„²π‘₯(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
4 nfcv 2904 . . . . . 6 β„²π‘₯𝑀
52, 3, 4nfov 7391 . . . . 5 β„²π‘₯(𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
61, 5nfmpt 5216 . . . 4 β„²π‘₯(𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
7 nfcv 2904 . . . 4 Ⅎ𝑀(𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
8 nfcv 2904 . . . . . . 7 Ⅎ𝑦𝑣
9 nfmpo1 7441 . . . . . . 7 Ⅎ𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
10 nfcv 2904 . . . . . . 7 Ⅎ𝑦𝑀
118, 9, 10nfov 7391 . . . . . 6 Ⅎ𝑦(𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
12 nfcv 2904 . . . . . 6 Ⅎ𝑣(𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀)
13 oveq1 7368 . . . . . 6 (𝑣 = 𝑦 β†’ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))
1411, 12, 13cbvmpt 5220 . . . . 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 7369 . . . . . 6 (𝑀 = π‘₯ β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀) = (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯))
1615mpteq2dv 5211 . . . . 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 2785 . . . 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 5220 . . 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 486 . . . . . 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 22965 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐽 ∈ (TopOnβ€˜π‘‹)) β†’ (𝐾 Γ—t 𝐽) ∈ (TopOnβ€˜(π‘Œ Γ— 𝑋)))
2421, 22, 23syl2anc 585 . . . . . . . . . . 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 22615 . . . . . . . . . . . . 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 22290 . . . . . . . . . . . 12 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
2927, 28sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
3022, 21, 25cnmptcom 23052 . . . . . . . . . . 11 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐿))
31 cnf2 22623 . . . . . . . . . . 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 1372 . . . . . . . . . 10 (πœ‘ β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴):(π‘Œ Γ— 𝑋)⟢βˆͺ 𝐿)
33 eqid 2733 . . . . . . . . . . 11 (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)
3433fmpo 8004 . . . . . . . . . 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 3233 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ π‘Œ) β†’ βˆ€π‘₯ ∈ 𝑋 𝐴 ∈ βˆͺ 𝐿)
3736r19.21bi 3233 . . . . . . 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 7506 . . . . . 6 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐿) β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = 𝐴)
4019, 20, 38, 39syl3anc 1372 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯) = 𝐴)
4140mpteq2dva 5209 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (𝑦 ∈ π‘Œ ↦ (𝑦(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)π‘₯)) = (𝑦 ∈ π‘Œ ↦ 𝐴))
4241mpteq2dva 5209 . . 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 2785 . 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 2733 . . . . 5 (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ βŸ¨π‘£, π‘€βŸ©)) = (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ βŸ¨π‘£, π‘€βŸ©))
4544xkoinjcn 23061 . . . 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 585 . . 3 (πœ‘ β†’ (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ βŸ¨π‘£, π‘€βŸ©)) ∈ (𝐽 Cn ((𝐾 Γ—t 𝐽) ↑ko 𝐾)))
4732feqmptd 6914 . . . 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 2835 . . 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 6846 . . . 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 7364 . . . 4 (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀) = ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜βŸ¨π‘£, π‘€βŸ©)
5149, 50eqtr4di 2791 . . 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 23055 . 2 (πœ‘ β†’ (𝑀 ∈ 𝑋 ↦ (𝑣 ∈ π‘Œ ↦ (𝑣(𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝐴)𝑀))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)))
5343, 52eqeltrrd 2835 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βŸ¨cop 4596  βˆͺ cuni 4869   ↦ cmpt 5192   Γ— cxp 5635  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Topctop 22265  TopOnctopon 22282   Cn ccn 22598   Γ—t ctx 22934   ↑ko cxko 22935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cn 22601  df-cnp 22602  df-cmp 22761  df-tx 22936  df-xko 22937
This theorem is referenced by:  xkocnv  23188  xkohmeo  23189
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