Theorem cncnp2m 14399

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 𝑦 ∈ 𝑋) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝑋 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

Step	Hyp	Ref	Expression
1		cntop1 14369	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2		cncnp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	toptopon 14186	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4	1, 3	sylib 122	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5		cntop2 14370	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6		cncnp.2	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
7	6	toptopon 14186	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
8	5, 7	sylib 122	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
9	2, 6	cnf 14372	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)
10	4, 8, 9	jca31 309	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
11	10	adantl 277	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
12	3	biimpi 120	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
13	12	3ad2ant1 1020	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
14	13	adantr 276	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
15	7	biimpi 120	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌))
16	15	3ad2ant2 1021	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
17	16	adantr 276	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝐾 ∈ (TopOn‘𝑌))
18		r19.2m 3533	. . . . . . 7 ⊢ ((∃𝑦 𝑦 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ∃𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
19	18	ex 115	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∃𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
20	19	3ad2ant3 1022	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∃𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
21		cnpf2 14375	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝐹:𝑋⟶𝑌)
22	21	3expia 1207	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋⟶𝑌))
23	22	rexlimdvw 2615	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (∃𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋⟶𝑌))
24	13, 16, 23	syl2anc 411	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) → (∃𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋⟶𝑌))
25	20, 24	syld 45	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋⟶𝑌))
26	25	imp 124	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → 𝐹:𝑋⟶𝑌)
27	14, 17, 26	jca31 309	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
28		cncnp 14398	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
29	28	baibd 924	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
30	11, 27, 29	pm5.21nd 917	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∃𝑦 𝑦 ∈ 𝑋) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ∪ cuni 3835  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  Topctop 14165  TopOnctopon 14178   Cn ccn 14353   CnP ccnp 14354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-topgen 12871  df-top 14166  df-topon 14179  df-cn 14356  df-cnp 14357

This theorem is referenced by: (None)