Theorem tgcn 23172

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 23155	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
5		tgcn.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = (topGen‘𝐵))
6		topontop 22833	. . . . . . . . . 10 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
7	2, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Top)
8	5, 7	eqeltrrd 2829	. . . . . . . 8 ⊢ (𝜑 → (topGen‘𝐵) ∈ Top)
9		tgclb 22890	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)
10	8, 9	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ TopBases)
11		bastg 22886	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (topGen‘𝐵))
13	12, 5	sseqtrrd 3981	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
14		ssralv 4012	. . . . 5 ⊢ (𝐵 ⊆ 𝐾 → (∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))
16	5	eleq2d 2814	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ 𝑥 ∈ (topGen‘𝐵)))
17		eltg3 22882	. . . . . . . . . 10 ⊢ (𝐵 ∈ TopBases → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧)))
18	10, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧)))
19	16, 18	bitrd 279	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐾 ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧)))
20		ssralv 4012	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
21		topontop 22833	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
22	1, 21	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Top)
23		iunopn 22818	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽) → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽)
24	23	ex 412	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → (∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
25	22, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
26	20, 25	sylan9r 508	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
27		imaeq2 6016	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ ∪ 𝑧))
28		imauni 7202	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ ∪ 𝑧) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦)
29	27, 28	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∪ 𝑧 → (◡𝐹 “ 𝑥) = ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦))
30	29	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑥 = ∪ 𝑧 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ∪ 𝑦 ∈ 𝑧 (◡𝐹 “ 𝑦) ∈ 𝐽))
31	30	imbi2d 340	. . . . . . . . . . 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 247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → (𝑥 = ∪ 𝑧 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
33	32	expimpd 453	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
34	33	exlimdv 1933	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧(𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
35	19, 34	sylbid 240	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐾 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽)))
36	35	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → (◡𝐹 “ 𝑥) ∈ 𝐽))
37	36	ralrimdva 3133	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))
38		imaeq2 6016	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑦))
39	38	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((◡𝐹 “ 𝑥) ∈ 𝐽 ↔ (◡𝐹 “ 𝑦) ∈ 𝐽))
40	39	cbvralvw 3213	. . . . 5 ⊢ (∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)
41	37, 40	imbitrdi 251	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽 → ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))
42	15, 41	impbid 212	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽))
43	42	anbi2d 630	. 2 ⊢ (𝜑 → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽)))
44	4, 43	bitrd 279	1 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐵 (◡𝐹 “ 𝑦) ∈ 𝐽)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3911  ∪ cuni 4867  ∪ ciun 4951  ^◡ccnv 5630   “ cima 5634  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  topGenctg 17376  Topctop 22813  TopOnctopon 22830  TopBasesctb 22865   Cn ccn 23144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-topgen 17382  df-top 22814  df-topon 22831  df-bases 22866  df-cn 23147

This theorem is referenced by:  subbascn  23174  txcnmpt  23544  ismtyhmeolem  37791