Theorem tgcn 14931

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 14920	. . 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 14737	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
7	2, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
8	5, 7	eqeltrrd 2309	. . . . . . . 8 ⊢ (𝜑 → (topGen‘𝐵) ∈ Top)
9		tgclb 14788	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
10	8, 9	sylibr 134	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ TopBases)
11		bastg 14784	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
12	10, 11	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (topGen‘𝐵))
13	12, 5	sseqtrrd 3266	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
14		ssralv 3291	. . . . 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 2301	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ 𝑥 ∈ (topGen‘𝐵)))
17		eltg3 14780	. . . . . . . . . 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 3291	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
21		topontop 14737	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
22	1, 21	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
23		iunopn 14725	. . . . . . . . . . . . . 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 5072	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ ∪ 𝑧))
28		imauni 5901	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ ∪ 𝑧) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦)
29	27, 28	eqtrdi 2280	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦))
30	29	eleq1d 2300	. . . . . . . . . . . 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 1867	. . . . . . . 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 2612	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
38		imaeq2 5072	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑦))
39	38	eleq1d 2300	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ 𝑦) ∈ 𝐽))
40	39	cbvralv 2767	. . . . 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 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510   ⊆ wss 3200  ∪ cuni 3893  ∪ ciun 3970  ^◡ccnv 4724   “ cima 4728  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017  topGenctg 13336  Topctop 14720  TopOnctopon 14733  TopBasesctb 14765   Cn ccn 14908

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-topgen 13342  df-top 14721  df-topon 14734  df-bases 14766  df-cn 14911

This theorem is referenced by:  txcnmpt  14996