Theorem cncnp2 23399

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 23358	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
2		cncnp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	toptopon 23035	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
4	1, 3	sylib 221	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
5		cntop2 23359	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6		cncnp.2	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
7	6	toptopon 23035	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
8	5, 7	sylib 221	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
9	2, 6	cnf 23364	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌)
10	4, 8, 9	jca31 523	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
11	10	adantl 486	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
12		r19.2z 4456	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ∃𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
13		cnptop1 23360	. . . . . 6 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐽 ∈ Top)
14	13, 3	sylib 221	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐽 ∈ (TopOn‘𝑋))
15		cnptop2 23361	. . . . . 6 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐾 ∈ Top)
16	15, 7	sylib 221	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐾 ∈ (TopOn‘𝑌))
17	2, 6	cnpf 23365	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → 𝐹:𝑋⟶𝑌)
18	14, 16, 17	jca31 523	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
19	18	rexlimivw 3162	. . 3 ⊢ (∃𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
20	12, 19	syl 18	. 2 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) → ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌))
21		cncnp 23398	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
22	21	baibd 548	. 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 813	1 ⊢ (𝑋 ≠ ∅ → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  ∅c0 4288  ∪ cuni 4868  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  Topctop 23011  TopOnctopon 23028   Cn ccn 23342   CnP ccnp 23343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-topgen 17486  df-top 23012  df-topon 23029  df-cn 23345  df-cnp 23346

This theorem is referenced by: (None)