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Theorem cnmptkp 21531
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 (𝐿 ^ko 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 cnmptkp.b . . . . 5 (𝜑𝐵𝑌)
21adantr 480 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑌)
3 cnmptk1.k . . . . . . . 8 (𝜑𝐾 ∈ (TopOn‘𝑌))
43adantr 480 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
5 cnmptk1.l . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘𝑍))
6 topontop 20766 . . . . . . . . . 10 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
75, 6syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ Top)
87adantr 480 . . . . . . . 8 ((𝜑𝑥𝑋) → 𝐿 ∈ Top)
9 eqid 2651 . . . . . . . . 9 𝐿 = 𝐿
109toptopon 20770 . . . . . . . 8 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
118, 10sylib 208 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐿 ∈ (TopOn‘ 𝐿))
12 cnmptk1.j . . . . . . . . . 10 (𝜑𝐽 ∈ (TopOn‘𝑋))
13 topontop 20766 . . . . . . . . . . . 12 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
143, 13syl 17 . . . . . . . . . . 11 (𝜑𝐾 ∈ Top)
15 eqid 2651 . . . . . . . . . . . 12 (𝐿 ^ko 𝐾) = (𝐿 ^ko 𝐾)
1615xkotopon 21451 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
1714, 7, 16syl2anc 694 . . . . . . . . . 10 (𝜑 → (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)))
18 cnmptkp.a . . . . . . . . . 10 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
19 cnf2 21101 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ^ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾))) → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2012, 17, 18, 19syl3anc 1366 . . . . . . . . 9 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
21 eqid 2651 . . . . . . . . . 10 (𝑥𝑋 ↦ (𝑦𝑌𝐴)) = (𝑥𝑋 ↦ (𝑦𝑌𝐴))
2221fmpt 6421 . . . . . . . . 9 (∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿) ↔ (𝑥𝑋 ↦ (𝑦𝑌𝐴)):𝑋⟶(𝐾 Cn 𝐿))
2320, 22sylibr 224 . . . . . . . 8 (𝜑 → ∀𝑥𝑋 (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
2423r19.21bi 2961 . . . . . . 7 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿))
25 cnf2 21101 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌𝐴) ∈ (𝐾 Cn 𝐿)) → (𝑦𝑌𝐴):𝑌 𝐿)
264, 11, 24, 25syl3anc 1366 . . . . . 6 ((𝜑𝑥𝑋) → (𝑦𝑌𝐴):𝑌 𝐿)
27 eqid 2651 . . . . . . 7 (𝑦𝑌𝐴) = (𝑦𝑌𝐴)
2827fmpt 6421 . . . . . 6 (∀𝑦𝑌 𝐴 𝐿 ↔ (𝑦𝑌𝐴):𝑌 𝐿)
2926, 28sylibr 224 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 𝐿)
30 cnmptkp.c . . . . . . 7 (𝑦 = 𝐵𝐴 = 𝐶)
3130eleq1d 2715 . . . . . 6 (𝑦 = 𝐵 → (𝐴 𝐿𝐶 𝐿))
3231rspcv 3336 . . . . 5 (𝐵𝑌 → (∀𝑦𝑌 𝐴 𝐿𝐶 𝐿))
332, 29, 32sylc 65 . . . 4 ((𝜑𝑥𝑋) → 𝐶 𝐿)
3430, 27fvmptg 6319 . . . 4 ((𝐵𝑌𝐶 𝐿) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
352, 33, 34syl2anc 694 . . 3 ((𝜑𝑥𝑋) → ((𝑦𝑌𝐴)‘𝐵) = 𝐶)
3635mpteq2dva 4777 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) = (𝑥𝑋𝐶))
37 toponuni 20767 . . . . . 6 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
383, 37syl 17 . . . . 5 (𝜑𝑌 = 𝐾)
391, 38eleqtrd 2732 . . . 4 (𝜑𝐵 𝐾)
40 eqid 2651 . . . . 5 𝐾 = 𝐾
4140xkopjcn 21507 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝐵 𝐾) → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤𝐵)) ∈ ((𝐿 ^ko 𝐾) Cn 𝐿))
4214, 7, 39, 41syl3anc 1366 . . 3 (𝜑 → (𝑤 ∈ (𝐾 Cn 𝐿) ↦ (𝑤𝐵)) ∈ ((𝐿 ^ko 𝐾) Cn 𝐿))
43 fveq1 6228 . . 3 (𝑤 = (𝑦𝑌𝐴) → (𝑤𝐵) = ((𝑦𝑌𝐴)‘𝐵))
4412, 18, 17, 42, 43cnmpt11 21514 . 2 (𝜑 → (𝑥𝑋 ↦ ((𝑦𝑌𝐴)‘𝐵)) ∈ (𝐽 Cn 𝐿))
4536, 44eqeltrrd 2731 1 (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941   cuni 4468  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  Topctop 20746  TopOnctopon 20763   Cn ccn 21076   ^ko cxko 21412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-pt 16152  df-top 20747  df-topon 20764  df-bases 20798  df-cn 21079  df-cmp 21238  df-xko 21414
This theorem is referenced by: (None)
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