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Theorem cnmptkp 22289
Description: The evaluation of the inner function in a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptk1.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptk1.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptk1.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmptkp.a (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
cnmptkp.b (𝜑𝐵𝑌)
cnmptkp.c (𝑦 = 𝐵𝐴 = 𝐶)
Assertion
Ref Expression
cnmptkp (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑍,𝑦   𝑥,𝐵   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑦,𝐵   𝑦,𝐶
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑥)

Proof of Theorem cnmptkp
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqid 2801 . . . 4 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2 cnmptkp.c . . . 4 (𝑦 = 𝐵𝐴 = 𝐶)
3 cnmptkp.b . . . . 5 (𝜑𝐵𝑌)
43adantr 484 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑌)
52eleq1d 2877 . . . . 5 (𝑦 = 𝐵 → (𝐴 𝐿𝐶 𝐿))
6 cnmptk1.k . . . . . . . 8 (𝜑𝐾 ∈ (TopOn‘𝑌))
76adantr 484 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
8 cnmptk1.l . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘𝑍))
9 topontop 21522 . . . . . . . . . 10 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
1110adantr 484 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝐿 ∈ Top)
12 toptopon2 21527 . . . . . . . 8 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1311, 12sylib 221 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘ 𝐿))
14 cnmptk1.j . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘𝑋))
15 topontop 21522 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
166, 15syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
17 eqid 2801 . . . . . . . . . . 11 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1817xkotopon 22209 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
1916, 10, 18syl2anc 587 . . . . . . . . 9 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
20 cnmptkp.a . . . . . . . . 9 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
21 cnf2 21858 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2214, 19, 20, 21syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2322fvmptelrn 6858 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
24 cnf2 21858 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌 𝐿)
257, 13, 23, 24syl3anc 1368 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌 𝐿)
261fmpt 6855 . . . . . 6 (∀𝑦𝑌 𝐴 𝐿 ↔ (𝑦𝑌𝐴):𝑌 𝐿)
2725, 26sylibr 237 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 𝐿)
285, 27, 4rspcdva 3576 . . . 4 ((𝜑𝑥𝑋) → 𝐶 𝐿)
291, 2, 4, 28fvmptd3 6772 . . 3 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
3029mpteq2dva 5128 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) = (𝑥𝑋𝐶))
31 toponuni 21523 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
326, 31syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
333, 32eleqtrd 2895 . . . 4 (𝜑𝐵 𝐾)
34 eqid 2801 . . . . 5 𝐾 = 𝐾
3534xkopjcn 22265 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝐵 𝐾) → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤𝐵)) ∈ ((𝐿ko 𝐾) Cn 𝐿))
3616, 10, 33, 35syl3anc 1368 . . 3 (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤𝐵)) ∈ ((𝐿ko 𝐾) Cn 𝐿))
37 fveq1 6648 . . 3 (𝑤 = (𝑦𝑌𝐴) → (𝑤𝐵) = ((𝑦𝑌𝐴)‘𝐵))
3814, 20, 19, 36, 37cnmpt11 22272 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
3930, 38eqeltrrd 2894 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109   cuni 4803  cmpt 5113  wf 6324  cfv 6328  (class class class)co 7139  Topctop 21502  TopOnctopon 21519   Cn ccn 21833  ko cxko 22170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-rest 16692  df-topgen 16713  df-pt 16714  df-top 21503  df-topon 21520  df-bases 21555  df-cn 21836  df-cmp 21996  df-xko 22172
This theorem is referenced by: (None)
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