Theorem tgcn 14651

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 14640	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
4	1, 2, 3	syl2anc 411	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
5		tgcn.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = (topGen‘𝐵))
6		topontop 14457	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
7	2, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
8	5, 7	eqeltrrd 2282	. . . . . . . 8 ⊢ (𝜑 → (topGen‘𝐵) ∈ Top)
9		tgclb 14508	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
10	8, 9	sylibr 134	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ TopBases)
11		bastg 14504	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
12	10, 11	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (topGen‘𝐵))
13	12, 5	sseqtrrd 3231	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
14		ssralv 3256	. . . . 5 ⊢ (𝐵 ⊆ 𝐾 → (∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))
15	13, 14	syl 14	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))
16	5	eleq2d 2274	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ 𝑥 ∈ (topGen‘𝐵)))
17		eltg3 14500	. . . . . . . . . 10 ⊢ (𝐵 ∈ TopBases → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧)))
18	10, 17	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧)))
19	16, 18	bitrd 188	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧)))
20		ssralv 3256	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
21		topontop 14457	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
22	1, 21	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
23		iunopn 14445	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽) → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽)
24	23	ex 115	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → (∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
25	22, 24	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
26	20, 25	sylan9r 410	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
27		imaeq2 5017	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ ∪ 𝑧))
28		imauni 5829	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ ∪ 𝑧) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦)
29	27, 28	eqtrdi 2253	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦))
30	29	eleq1d 2273	. . . . . . . . . . . 12 ⊢ (𝑥 = ∪ 𝑧 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
31	30	imbi2d 230	. . . . . . . . . . 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 157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → (𝑥 = ∪ 𝑧 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
33	32	expimpd 363	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
34	33	exlimdv 1841	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
35	19, 34	sylbid 150	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐾 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
36	35	imp 124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽))
37	36	ralrimdva 2585	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
38		imaeq2 5017	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑦))
39	38	eleq1d 2273	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ 𝑦) ∈ 𝐽))
40	39	cbvralv 2737	. . . . 5 ⊢ (∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)
41	37, 40	imbitrdi 161	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))
42	15, 41	impbid 129	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))
43	42	anbi2d 464	. 2 ⊢ (𝜑 → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽)))
44	4, 43	bitrd 188	1 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372  ∃wex 1514   ∈ wcel 2175  ∀wral 2483   ⊆ wss 3165  ∪ cuni 3849  ∪ ciun 3926  ^◡ccnv 4673   “ cima 4677  ⟶wf 5266  ‘cfv 5270  (class class class)co 5943  topGenctg 13057  Topctop 14440  TopOnctopon 14453  TopBasesctb 14485   Cn ccn 14628

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-map 6736  df-topgen 13063  df-top 14441  df-topon 14454  df-bases 14486  df-cn 14631

This theorem is referenced by:  txcnmpt  14716