Theorem tgcn 23439

Description: The continuity predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 7-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))

Assertion

Ref	Expression
tgcn	⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽)))

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

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem tgcn

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgcn.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		tgcn.4	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		iscn 23422	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
4	1, 2, 3	syl2anc 596	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
5		tgcn.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = (topGen‘𝐵))
6		topontop 23100	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
7	2, 6	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
8	5, 7	eqeltrrd 2867	. . . . . . . 8 ⊢ (𝜑 → (topGen‘𝐵) ∈ Top)
9		tgclb 23157	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
10	8, 9	sylibr 237	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ TopBases)
11		bastg 23153	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
12	10, 11	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (topGen‘𝐵))
13	12, 5	sseqtrrd 3977	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
14		ssralv 4009	. . . . 5 ⊢ (𝐵 ⊆ 𝐾 → (∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))
15	13, 14	syl 18	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))
16	5	eleq2d 2852	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ 𝑥 ∈ (topGen‘𝐵)))
17		eltg3 23149	. . . . . . . . . 10 ⊢ (𝐵 ∈ TopBases → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧)))
18	10, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧)))
19	16, 18	bitrd 282	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧)))
20		ssralv 4009	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
21		topontop 23100	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
22	1, 21	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
23		iunopn 23085	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽) → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽)
24	23	ex 418	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → (∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
25	22, 24	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
26	20, 25	sylan9r 518	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
27		imaeq2 6063	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ ∪ 𝑧))
28		imauni 7251	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ ∪ 𝑧) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦)
29	27, 28	eqtrdi 2817	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦))
30	29	eleq1d 2851	. . . . . . . . . . . 12 ⊢ (𝑥 = ∪ 𝑧 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
31	30	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ 𝑧 → ((∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽) ↔ (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽)))
32	26, 31	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → (𝑥 = ∪ 𝑧 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
33	32	expimpd 459	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
34	33	exlimdv 1966	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
35	19, 34	sylbid 243	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐾 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
36	35	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽))
37	36	ralrimdva 3168	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
38		imaeq2 6063	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑦))
39	38	eleq1d 2851	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ 𝑦) ∈ 𝐽))
40	39	cbvralvw 3246	. . . . 5 ⊢ (∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)
41	37, 40	imbitrdi 254	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))
42	15, 41	impbid 215	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))
43	42	anbi2d 642	. 2 ⊢ (𝜑 → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽)))
44	4, 43	bitrd 282	1 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082   ⊆ wss 3908  ∪ cuni 4877  ∪ ciun 4961  ^◡ccnv 5665   “ cima 5669  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  topGenctg 17515  Topctop 23080  TopOnctopon 23097  TopBasesctb 23132   Cn ccn 23411

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8835  df-topgen 17521  df-top 23081  df-topon 23098  df-bases 23133  df-cn 23414

This theorem is used by:  subbascn  23441  txcnmpt  23811  ismtyhmeolem  38488