ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cncnp2m GIF version

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)
  Copyright terms: Public domain W3C validator