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Theorem cncnp2 21413
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 21372 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2 cncnp.1 . . . . . 6 𝑋 = 𝐽
32toptopon 21049 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
41, 3sylib 210 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5 cntop2 21373 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6 cncnp.2 . . . . . 6 𝑌 = 𝐾
76toptopon 21049 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
85, 7sylib 210 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
92, 6cnf 21378 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌)
104, 8, 9jca31 511 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
1110adantl 474 . 2 ((𝑋 ≠ ∅ ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
12 r19.2z 4254 . . 3 ((𝑋 ≠ ∅ ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ∃𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
13 cnptop1 21374 . . . . . 6 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐽 ∈ Top)
1413, 3sylib 210 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐽 ∈ (TopOn‘𝑋))
15 cnptop2 21375 . . . . . 6 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐾 ∈ Top)
1615, 7sylib 210 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐾 ∈ (TopOn‘𝑌))
172, 6cnpf 21379 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋𝑌)
1814, 16, 17jca31 511 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
1918rexlimivw 3211 . . 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 21412 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2221baibd 536 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
2311, 20, 22pm5.21nd 837 1 (𝑋 ≠ ∅ → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wne 2972  wral 3090  wrex 3091  c0 4116   cuni 4629  wf 6098  cfv 6102  (class class class)co 6879  Topctop 21025  TopOnctopon 21042   Cn ccn 21356   CnP ccnp 21357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1st 7402  df-2nd 7403  df-map 8098  df-topgen 16418  df-top 21026  df-topon 21043  df-cn 21359  df-cnp 21360
This theorem is referenced by: (None)
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