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Theorem cncnp2 21079
 Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
cncnp.1 𝑋 = 𝐽
cncnp.2 𝑌 = 𝐾
Assertion
Ref Expression
cncnp2 (𝑋 ≠ ∅ → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑥,𝑌

Proof of Theorem cncnp2
StepHypRef Expression
1 cntop1 21038 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2 cncnp.1 . . . . . 6 𝑋 = 𝐽
32toptopon 20716 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
41, 3sylib 208 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5 cntop2 21039 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6 cncnp.2 . . . . . 6 𝑌 = 𝐾
76toptopon 20716 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
85, 7sylib 208 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
92, 6cnf 21044 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌)
104, 8, 9jca31 557 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
1110adantl 482 . 2 ((𝑋 ≠ ∅ ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
12 r19.2z 4058 . . 3 ((𝑋 ≠ ∅ ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ∃𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
13 cnptop1 21040 . . . . . 6 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐽 ∈ Top)
1413, 3sylib 208 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐽 ∈ (TopOn‘𝑋))
15 cnptop2 21041 . . . . . 6 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐾 ∈ Top)
1615, 7sylib 208 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐾 ∈ (TopOn‘𝑌))
172, 6cnpf 21045 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋𝑌)
1814, 16, 17jca31 557 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
1918rexlimivw 3027 . . 3 (∃𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
2012, 19syl 17 . 2 ((𝑋 ≠ ∅ ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
21 cncnp 21078 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2221baibd 948 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
2311, 20, 22pm5.21nd 941 1 (𝑋 ≠ ∅ → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1482   ∈ wcel 1989   ≠ wne 2793  ∀wral 2911  ∃wrex 2912  ∅c0 3913  ∪ cuni 4434  ⟶wf 5882  ‘cfv 5886  (class class class)co 6647  Topctop 20692  TopOnctopon 20709   Cn ccn 21022   CnP ccnp 21023 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-map 7856  df-topgen 16098  df-top 20693  df-topon 20710  df-cn 21025  df-cnp 21026 This theorem is referenced by: (None)
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