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Theorem cncnp2m 12430
 Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.)
Hypotheses
Ref Expression
cncnp.1 𝑋 = 𝐽
cncnp.2 𝑌 = 𝐾
Assertion
Ref Expression
cncnp2m ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑦,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐹(𝑦)   𝐽(𝑦)   𝐾(𝑦)   𝑌(𝑦)

Proof of Theorem cncnp2m
StepHypRef Expression
1 cntop1 12400 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2 cncnp.1 . . . . . 6 𝑋 = 𝐽
32toptopon 12215 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
41, 3sylib 121 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5 cntop2 12401 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6 cncnp.2 . . . . . 6 𝑌 = 𝐾
76toptopon 12215 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
85, 7sylib 121 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
92, 6cnf 12403 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋𝑌)
104, 8, 9jca31 307 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
1110adantl 275 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
123biimpi 119 . . . . 5 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
13123ad2ant1 1003 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) → 𝐽 ∈ (TopOn‘𝑋))
1413adantr 274 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
157biimpi 119 . . . . 5 (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌))
16153ad2ant2 1004 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) → 𝐾 ∈ (TopOn‘𝑌))
1716adantr 274 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝐾 ∈ (TopOn‘𝑌))
18 r19.2m 3450 . . . . . . 7 ((∃𝑦 𝑦𝑋 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ∃𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
1918ex 114 . . . . . 6 (∃𝑦 𝑦𝑋 → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∃𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
20193ad2ant3 1005 . . . . 5 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∃𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
21 cnpf2 12406 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝐹:𝑋𝑌)
22213expia 1184 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋𝑌))
2322rexlimdvw 2554 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (∃𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋𝑌))
2413, 16, 23syl2anc 409 . . . . 5 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) → (∃𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋𝑌))
2520, 24syld 45 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋𝑌))
2625imp 123 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝐹:𝑋𝑌)
2714, 17, 26jca31 307 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌))
28 cncnp 12429 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2928baibd 909 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
3011, 27, 29pm5.21nd 902 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦𝑋) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  ∪ cuni 3740  ⟶wf 5123  ‘cfv 5127  (class class class)co 5778  Topctop 12194  TopOnctopon 12207   Cn ccn 12384   CnP ccnp 12385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-map 6548  df-topgen 12171  df-top 12195  df-topon 12208  df-cn 12387  df-cnp 12388 This theorem is referenced by: (None)
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