Theorem cncnp2 23175

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) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑥,𝑌

Proof of Theorem cncnp2

Step	Hyp	Ref	Expression
1		cntop1 23134	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2		cncnp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	toptopon 22811	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4	1, 3	sylib 218	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5		cntop2 23135	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6		cncnp.2	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
7	6	toptopon 22811	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
8	5, 7	sylib 218	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
9	2, 6	cnf 23140	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)
10	4, 8, 9	jca31 514	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
11	10	adantl 481	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
12		r19.2z 4461	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ∃𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
13		cnptop1 23136	. . . . . 6 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐽 ∈ Top)
14	13, 3	sylib 218	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐽 ∈ (TopOn‘𝑋))
15		cnptop2 23137	. . . . . 6 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐾 ∈ Top)
16	15, 7	sylib 218	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐾 ∈ (TopOn‘𝑌))
17	2, 6	cnpf 23141	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋⟶𝑌)
18	14, 16, 17	jca31 514	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
19	18	rexlimivw 3131	. . 3 ⊢ (∃𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
20	12, 19	syl 17	. 2 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
21		cncnp 23174	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
22	21	baibd 539	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
23	11, 20, 22	pm5.21nd 801	1 ⊢ (𝑋 ≠ ∅ → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  ∅c0 4299  ∪ cuni 4874  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Topctop 22787  TopOnctopon 22804   Cn ccn 23118   CnP ccnp 23119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-topgen 17413  df-top 22788  df-topon 22805  df-cn 23121  df-cnp 23122

This theorem is referenced by: (None)