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Theorem cnmptkp 23704
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 2735 . . . 4 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2 cnmptkp.c . . . 4 (𝑦 = 𝐵𝐴 = 𝐶)
3 cnmptkp.b . . . . 5 (𝜑𝐵𝑌)
43adantr 480 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑌)
52eleq1d 2824 . . . . 5 (𝑦 = 𝐵 → (𝐴 𝐿𝐶 𝐿))
6 cnmptk1.k . . . . . . . 8 (𝜑𝐾 ∈ (TopOn‘𝑌))
76adantr 480 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
8 cnmptk1.l . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘𝑍))
9 topontop 22935 . . . . . . . . . 10 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
108, 9syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
1110adantr 480 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝐿 ∈ Top)
12 toptopon2 22940 . . . . . . . 8 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1311, 12sylib 218 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘ 𝐿))
14 cnmptk1.j . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘𝑋))
15 topontop 22935 . . . . . . . . . . 11 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
166, 15syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
17 eqid 2735 . . . . . . . . . . 11 (𝐿ko 𝐾) = (𝐿ko 𝐾)
1817xkotopon 23624 . . . . . . . . . 10 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
1916, 10, 18syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐿ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
20 cnmptkp.a . . . . . . . . 9 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
21 cnf2 23273 . . . . . . . . 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 1370 . . . . . . . 8 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2322fvmptelcdm 7133 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
24 cnf2 23273 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌 𝐿)
257, 13, 23, 24syl3anc 1370 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌 𝐿)
261fmpt 7130 . . . . . 6 (∀𝑦𝑌 𝐴 𝐿 ↔ (𝑦𝑌𝐴):𝑌 𝐿)
2725, 26sylibr 234 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 𝐿)
285, 27, 4rspcdva 3623 . . . 4 ((𝜑𝑥𝑋) → 𝐶 𝐿)
291, 2, 4, 28fvmptd3 7039 . . 3 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
3029mpteq2dva 5248 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) = (𝑥𝑋𝐶))
31 toponuni 22936 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
326, 31syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
333, 32eleqtrd 2841 . . . 4 (𝜑𝐵 𝐾)
34 eqid 2735 . . . . 5 𝐾 = 𝐾
3534xkopjcn 23680 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝐵 𝐾) → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤𝐵)) ∈ ((𝐿ko 𝐾) Cn 𝐿))
3616, 10, 33, 35syl3anc 1370 . . 3 (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤𝐵)) ∈ ((𝐿ko 𝐾) Cn 𝐿))
37 fveq1 6906 . . 3 (𝑤 = (𝑦𝑌𝐴) → (𝑤𝐵) = ((𝑦𝑌𝐴)‘𝐵))
3814, 20, 19, 36, 37cnmpt11 23687 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
3930, 38eqeltrrd 2840 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059   cuni 4912  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  ko cxko 23585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-pt 17491  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-cmp 23411  df-xko 23587
This theorem is referenced by: (None)
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