Theorem tgcnp 23078

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 23062	. . . 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 22736	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
8	2, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
9	6, 8	eqeltrrd 2833	. . . . . . . 8 ⊢ (𝜑 → (topGen‘𝐵) ∈ Top)
10		tgclb 22794	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
11	9, 10	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ TopBases)
12		bastg 22790	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (topGen‘𝐵))
14	13, 6	sseqtrrd 4023	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
15		ssralv 4050	. . . . 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 611	. . 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 2818	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝐾 ↔ 𝑧 ∈ (topGen‘𝐵)))
20	19	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐾) → 𝑧 ∈ (topGen‘𝐵))
21		tg2 22789	. . . . . . . . 9 ⊢ ((𝑧 ∈ (topGen‘𝐵) ∧ (𝐹‘𝑃) ∈ 𝑧) → ∃𝑦 ∈ 𝐵 ((𝐹‘𝑃) ∈ 𝑦 ∧ 𝑦 ⊆ 𝑧))
22		r19.29 3113	. . . . . . . . . . 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 3990	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 “ 𝑥) ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑧) → (𝐹 “ 𝑥) ⊆ 𝑧)
24	23	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ⊆ 𝑧 → ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ 𝑥) ⊆ 𝑧))
25	24	anim2d 611	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝑧 → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑧)))
26	25	reximdv 3169	. . . . . . . . . . . . . . 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 418	. . . . . . . . . . . 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 3150	. . . . . . . . . . 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 413	. . . . . . . . 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 412	. . . . . . 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 3153	. . . 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 611	. . 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 23062	. . . 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 259	. 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 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069   ⊆ wss 3948   “ cima 5679  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412  topGenctg 17390  Topctop 22716  TopOnctopon 22733  TopBasesctb 22769   CnP ccnp 23050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-topgen 17396  df-top 22717  df-topon 22734  df-bases 22770  df-cnp 23053

This theorem is referenced by:  txcnp  23445  ptcnp  23447  metcnp3  24370