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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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