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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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