Theorem tgcnp 22404

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 22388	. . . 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 1370	. . 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 22062	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
8	2, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
9	6, 8	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜑 → (topGen‘𝐵) ∈ Top)
10		tgclb 22120	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
11	9, 10	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ TopBases)
12		bastg 22116	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (topGen‘𝐵))
14	13, 6	sseqtrrd 3962	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
15		ssralv 3987	. . . . 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 17	. . . 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 612	. . 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 239	. 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 2824	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝐾 ↔ 𝑧 ∈ (topGen‘𝐵)))
20	19	biimpa 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐾) → 𝑧 ∈ (topGen‘𝐵))
21		tg2 22115	. . . . . . . . 9 ⊢ ((𝑧 ∈ (topGen‘𝐵) ∧ (𝐹‘𝑃) ∈ 𝑧) → ∃𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))
22		r19.29 3184	. . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 “ 𝑥) ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑧) → (𝐹 “ 𝑥) ⊆ 𝑧)
24	23	expcom 414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ⊆ 𝑧 → ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ 𝑥) ⊆ 𝑧))
25	24	anim2d 612	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝑧 → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))
26	25	reximdv 3202	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ 𝑧 → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))
27	26	com12 32	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑦 ⊆ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))
28	27	imim2i 16	. . . . . . . . . . . . 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 419	. . . . . . . . . . . 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 3211	. . . . . . . . . . 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 17	. . . . . . . . . 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 414	. . . . . . . . 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 17	. . . . . . . 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 413	. . . . . . 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 86	. . . . . 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 17	. . . . 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 3106	. . . 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 612	. . 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 22388	. . . 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 1370	. . 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 258	. 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 211	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   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887   “ cima 5592  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  topGenctg 17148  Topctop 22042  TopOnctopon 22059  TopBasesctb 22095   CnP ccnp 22376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cnp 22379

This theorem is referenced by:  txcnp  22771  ptcnp  22773  metcnp3  23696