Theorem tgcnp 14923

Description: The "continuous at a point" predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
tgcn.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
tgcn.3	⊢ (𝜑 → 𝐾 = (topGen‘𝐵))
tgcn.4	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
tgcnp.5	⊢ (𝜑 → 𝑃 ∈ 𝑋)

Assertion

Ref	Expression
tgcnp	⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝑃,𝑦   𝜑,𝑥   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem tgcnp

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgcn.1	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		tgcn.4	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		tgcnp.5	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
4		iscnp 14913	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
5	1, 2, 3, 4	syl3anc 1271	. . 3 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
6		tgcn.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = (topGen‘𝐵))
7		topontop 14728	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
8	2, 7	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
9	6, 8	eqeltrrd 2307	. . . . . . . 8 ⊢ (𝜑 → (topGen‘𝐵) ∈ Top)
10		tgclb 14779	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
11	9, 10	sylibr 134	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ TopBases)
12		bastg 14775	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
13	11, 12	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (topGen‘𝐵))
14	13, 6	sseqtrrd 3264	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
15		ssralv 3289	. . . . 5 ⊢ (𝐵 ⊆ 𝐾 → (∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))
16	14, 15	syl 14	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))))
17	16	anim2d 337	. . 3 ⊢ (𝜑 → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
18	5, 17	sylbid 150	. 2 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))
19	6	eleq2d 2299	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝐾 ↔ 𝑧 ∈ (topGen‘𝐵)))
20	19	biimpa 296	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐾) → 𝑧 ∈ (topGen‘𝐵))
21		tg2 14774	. . . . . . . . 9 ⊢ ((𝑧 ∈ (topGen‘𝐵) ∧ (𝐹‘𝑃) ∈ 𝑧) → ∃𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))
22		r19.29 2668	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ∧ ∃𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧)) → ∃𝑦 ∈ 𝐵 (((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧)))
23		sstr 3233	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 “ 𝑥) ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑧) → (𝐹 “ 𝑥) ⊆ 𝑧)
24	23	expcom 116	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ⊆ 𝑧 → ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ 𝑥) ⊆ 𝑧))
25	24	anim2d 337	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝑧 → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))
26	25	reximdv 2631	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ 𝑧 → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))
27	26	com12 30	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑦 ⊆ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))
28	27	imim2i 12	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ((𝐹‘𝑃) ∈ 𝑦 → (𝑦 ⊆ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧))))
29	28	imp32 257	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧))
30	29	rexlimivw 2644	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝐵 (((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧))
31	22, 30	syl 14	. . . . . . . . . 10 ⊢ ((∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) ∧ ∃𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧))
32	31	expcom 116	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))
33	21, 32	syl 14	. . . . . . . 8 ⊢ ((𝑧 ∈ (topGen‘𝐵) ∧ (𝐹‘𝑃) ∈ 𝑧) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))
34	33	ex 115	. . . . . . 7 ⊢ (𝑧 ∈ (topGen‘𝐵) → ((𝐹‘𝑃) ∈ 𝑧 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧))))
35	34	com23 78	. . . . . 6 ⊢ (𝑧 ∈ (topGen‘𝐵) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ((𝐹‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧))))
36	20, 35	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐾) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ((𝐹‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧))))
37	36	ralrimdva 2610	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)) → ∀𝑧 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧))))
38	37	anim2d 337	. . 3 ⊢ (𝜑 → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))))
39		iscnp 14913	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))))
40	1, 2, 3, 39	syl3anc 1271	. . 3 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑧 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))))
41	38, 40	sylibrd 169	. 2 ⊢ (𝜑 → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)))
42	18, 41	impbid 129	1 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   ⊆ wss 3198   “ cima 4726  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013  topGenctg 13327  Topctop 14711  TopOnctopon 14724  TopBasesctb 14756   CnP ccnp 14900

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-map 6814  df-topgen 13333  df-top 14712  df-topon 14725  df-bases 14757  df-cnp 14903

This theorem is referenced by:  txcnp  14985  metcnp3  15225