Theorem tgcn 14376

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 14365	. . 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 14182	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
7	2, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
8	5, 7	eqeltrrd 2271	. . . . . . . 8 ⊢ (𝜑 → (topGen‘𝐵) ∈ Top)
9		tgclb 14233	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
10	8, 9	sylibr 134	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ TopBases)
11		bastg 14229	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
12	10, 11	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (topGen‘𝐵))
13	12, 5	sseqtrrd 3218	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
14		ssralv 3243	. . . . 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 2263	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ 𝑥 ∈ (topGen‘𝐵)))
17		eltg3 14225	. . . . . . . . . 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 3243	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
21		topontop 14182	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
22	1, 21	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
23		iunopn 14170	. . . . . . . . . . . . . 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 5001	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ ∪ 𝑧))
28		imauni 5804	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ ∪ 𝑧) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦)
29	27, 28	eqtrdi 2242	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦))
30	29	eleq1d 2262	. . . . . . . . . . . 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 1830	. . . . . . . 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 2574	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
38		imaeq2 5001	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑦))
39	38	eleq1d 2262	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ 𝑦) ∈ 𝐽))
40	39	cbvralv 2726	. . . . 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 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3153  ∪ cuni 3835  ∪ ciun 3912  ^◡ccnv 4658   “ cima 4662  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  topGenctg 12865  Topctop 14165  TopOnctopon 14178  TopBasesctb 14210   Cn ccn 14353

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-topgen 12871  df-top 14166  df-topon 14179  df-bases 14211  df-cn 14356

This theorem is referenced by:  txcnmpt  14441