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Theorem cncnpi 20830
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 2605 . . . 4 𝐾 = 𝐾
31, 2cnf 20798 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
43adantr 479 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋 𝐾)
5 cnima 20817 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
65ad2ant2r 778 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → (𝐹𝑦) ∈ 𝐽)
7 simpr 475 . . . . . . 7 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐴𝑋)
87adantr 479 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → 𝐴𝑋)
9 simprr 791 . . . . . 6 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → (𝐹𝐴) ∈ 𝑦)
103ad2antrr 757 . . . . . . 7 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → 𝐹:𝑋 𝐾)
11 ffn 5940 . . . . . . 7 (𝐹:𝑋 𝐾𝐹 Fn 𝑋)
12 elpreima 6226 . . . . . . 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 958 . . . . 5 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → 𝐴 ∈ (𝐹𝑦))
15 eqimss 3615 . . . . . . . 8 (𝑥 = (𝐹𝑦) → 𝑥 ⊆ (𝐹𝑦))
1615biantrud 526 . . . . . . 7 (𝑥 = (𝐹𝑦) → (𝐴𝑥 ↔ (𝐴𝑥𝑥 ⊆ (𝐹𝑦))))
17 eleq2 2672 . . . . . . 7 (𝑥 = (𝐹𝑦) → (𝐴𝑥𝐴 ∈ (𝐹𝑦)))
1816, 17bitr3d 268 . . . . . 6 (𝑥 = (𝐹𝑦) → ((𝐴𝑥𝑥 ⊆ (𝐹𝑦)) ↔ 𝐴 ∈ (𝐹𝑦)))
1918rspcev 3277 . . . . 5 (((𝐹𝑦) ∈ 𝐽𝐴 ∈ (𝐹𝑦)) → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦)))
206, 14, 19syl2anc 690 . . . 4 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝐴) ∈ 𝑦)) → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦)))
2120expr 640 . . 3 (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) ∧ 𝑦𝐾) → ((𝐹𝐴) ∈ 𝑦 → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦))))
2221ralrimiva 2944 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → ∀𝑦𝐾 ((𝐹𝐴) ∈ 𝑦 → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦))))
23 cntop1 20792 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2423adantr 479 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐽 ∈ Top)
251toptopon 20486 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
2624, 25sylib 206 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27 cntop2 20793 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
2827adantr 479 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐾 ∈ Top)
292toptopon 20486 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3028, 29sylib 206 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐾 ∈ (TopOn‘ 𝐾))
31 iscnp3 20796 . . 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 1317 . 2 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋 𝐾 ∧ ∀𝑦𝐾 ((𝐹𝐴) ∈ 𝑦 → ∃𝑥𝐽 (𝐴𝑥𝑥 ⊆ (𝐹𝑦))))))
334, 22, 32mpbir2and 958 1 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  wral 2891  wrex 2892  wss 3535   cuni 4362  ccnv 5023  cima 5027   Fn wfn 5781  wf 5782  cfv 5786  (class class class)co 6523  Topctop 20455  TopOnctopon 20456   Cn ccn 20776   CnP ccnp 20777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-map 7719  df-top 20459  df-topon 20461  df-cn 20779  df-cnp 20780
This theorem is referenced by:  cnsscnp  20831  cncnp  20832  lmcn  20857  ptcn  21178  tmdcn2  21641  ghmcnp  21666  tsmsmhm  21697  tsmsadd  21698  dvcnp2  23402  dvaddbr  23420  dvmulbr  23421  dvcobr  23428  dvcjbr  23431  dvcnvlem  23456  lhop1lem  23493  dvcnvrelem2  23498  ftc1cn  23523  taylthlem2  23845  psercn  23897  abelth  23912  cxpcn3  24202  efrlim  24409  blocni  26846  cvmlift2lem11  30351  cvmlift2lem12  30352  cvmlift3lem7  30363  poimir  32411  ftc1cnnc  32453  cncfiooicclem1  38579  fouriercn  38925
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