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Theorem cncnpi 21303
Description: A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
cnsscnp.1 𝑋 = 𝐽
Assertion
Ref Expression
cncnpi ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))

Proof of Theorem cncnpi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnsscnp.1 . . . 4 𝑋 = 𝐽
2 eqid 2771 . . . 4 𝐾 = 𝐾
31, 2cnf 21271 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
43adantr 466 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋 𝐾)
5 cnima 21290 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
65ad2ant2r 741 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → (𝐹𝑦) ∈ 𝐽)
7 simpr 471 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐴𝑋)
87adantr 466 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → 𝐴𝑋)
9 simprr 756 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → (𝐹𝐴) ∈ 𝑦)
103ad2antrr 705 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → 𝐹:𝑋 𝐾)
11 ffn 6185 . . . . . . 7 (𝐹:𝑋 𝐾𝐹 Fn 𝑋)
12 elpreima 6480 . . . . . . 7 (𝐹 Fn 𝑋 → (𝐴 ∈ (𝐹𝑦) ↔ (𝐴𝑋 ∧ (𝐹𝐴) ∈ 𝑦)))
1310, 11, 123syl 18 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → (𝐴 ∈ (𝐹𝑦) ↔ (𝐴𝑋 ∧ (𝐹𝐴) ∈ 𝑦)))
148, 9, 13mpbir2and 692 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → 𝐴 ∈ (𝐹𝑦))
15 eqimss 3806 . . . . . . . 8 (𝑥 = (𝐹𝑦) → 𝑥 ⊆ (𝐹𝑦))
1615biantrud 521 . . . . . . 7 (𝑥 = (𝐹𝑦) → (𝐴𝑥 ↔ (𝐴𝑥𝑥 ⊆ (𝐹𝑦))))
17 eleq2 2839 . . . . . . 7 (𝑥 = (𝐹𝑦) → (𝐴𝑥𝐴 ∈ (𝐹𝑦)))
1816, 17bitr3d 270 . . . . . 6 (𝑥 = (𝐹𝑦) → ((𝐴𝑥𝑥 ⊆ (𝐹𝑦)) ↔ 𝐴 ∈ (𝐹𝑦)))
1918rspcev 3460 . . . . 5 (((𝐹𝑦) ∈ 𝐽𝐴 ∈ (𝐹𝑦)) → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦)))
206, 14, 19syl2anc 573 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦)))
2120expr 444 . . 3 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ 𝑦𝐾) → ((𝐹𝐴) ∈ 𝑦 → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦))))
2221ralrimiva 3115 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → ∀𝑦𝐾 ((𝐹𝐴) ∈ 𝑦 → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦))))
23 cntop1 21265 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2423adantr 466 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
251toptopon 20942 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
2624, 25sylib 208 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27 cntop2 21266 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2827adantr 466 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐾 ∈ Top)
292toptopon 20942 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3028, 29sylib 208 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐾 ∈ (TopOn‘ 𝐾))
31 iscnp3 21269 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋 𝐾 ∧ ∀𝑦𝐾 ((𝐹𝐴) ∈ 𝑦 → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦))))))
3226, 30, 7, 31syl3anc 1476 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋 𝐾 ∧ ∀𝑦𝐾 ((𝐹𝐴) ∈ 𝑦 → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦))))))
334, 22, 32mpbir2and 692 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  wss 3723   cuni 4574  ccnv 5248  cima 5252   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  Topctop 20918  TopOnctopon 20935   Cn ccn 21249   CnP ccnp 21250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-top 20919  df-topon 20936  df-cn 21252  df-cnp 21253
This theorem is referenced by:  cnsscnp  21304  cncnp  21305  lmcn  21330  ptcn  21651  tmdcn2  22113  ghmcnp  22138  tsmsmhm  22169  tsmsadd  22170  dvcnp2  23903  dvaddbr  23921  dvmulbr  23922  dvcobr  23929  dvcjbr  23932  dvcnvlem  23959  lhop1lem  23996  dvcnvrelem2  24001  ftc1cn  24026  taylthlem2  24348  psercn  24400  abelth  24415  cxpcn3  24710  efrlim  24917  blocni  28000  cvmlift2lem11  31633  cvmlift2lem12  31634  cvmlift3lem7  31645  poimir  33775  ftc1cnnc  33816  cncfiooicclem1  40624  fouriercn  40966
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