Theorem dissnref 23252

Description: The set of singletons is a refinement of any open covering of the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.)

Hypothesis

Ref	Expression
dissnref.c	⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}

Assertion

Ref	Expression
dissnref	⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → 𝐶Ref𝑌)

Distinct variable groups:   𝑢,𝐶,𝑥   𝑢,𝑉,𝑥   𝑢,𝑋,𝑥   𝑢,𝑌,𝑥

Proof of Theorem dissnref

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 483	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → ∪ 𝑌 = 𝑋)
2		dissnref.c	. . . 4 ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}
3	2	unisngl 23251	. . 3 ⊢ 𝑋 = ∪ 𝐶
4	1, 3	eqtrdi 2786	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → ∪ 𝑌 = ∪ 𝐶)
5		simplr 765	. . . . . 6 ⊢ ((((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑦)) → 𝑢 = {𝑥})
6		simprr 769	. . . . . . 7 ⊢ ((((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑦)) → 𝑥 ∈ 𝑦)
7	6	snssd 4811	. . . . . 6 ⊢ ((((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑦)) → {𝑥} ⊆ 𝑦)
8	5, 7	eqsstrd 4019	. . . . 5 ⊢ ((((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑦)) → 𝑢 ⊆ 𝑦)
9		simplr 765	. . . . . . 7 ⊢ (((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 ∈ 𝑋)
10		simp-4r 780	. . . . . . 7 ⊢ (((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → ∪ 𝑌 = 𝑋)
11	9, 10	eleqtrrd 2834	. . . . . 6 ⊢ (((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 ∈ ∪ 𝑌)
12		eluni2 4911	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑌 ↔ ∃𝑦 ∈ 𝑌 𝑥 ∈ 𝑦)
13	11, 12	sylib 217	. . . . 5 ⊢ (((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → ∃𝑦 ∈ 𝑌 𝑥 ∈ 𝑦)
14	8, 13	reximddv 3169	. . . 4 ⊢ (((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → ∃𝑦 ∈ 𝑌 𝑢 ⊆ 𝑦)
15	2	eqabri 2875	. . . . . 6 ⊢ (𝑢 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
16	15	biimpi 215	. . . . 5 ⊢ (𝑢 ∈ 𝐶 → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
17	16	adantl 480	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
18	14, 17	r19.29a 3160	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) → ∃𝑦 ∈ 𝑌 𝑢 ⊆ 𝑦)
19	18	ralrimiva 3144	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → ∀𝑢 ∈ 𝐶 ∃𝑦 ∈ 𝑌 𝑢 ⊆ 𝑦)
20		pwexg 5375	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
21		simpr 483	. . . . . . . . 9 ⊢ (((𝑢 ∈ 𝐶 ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
22		snelpwi 5442	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
23	22	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝑢 ∈ 𝐶 ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
24	21, 23	eqeltrd 2831	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐶 ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
25	24, 16	r19.29a 3160	. . . . . . 7 ⊢ (𝑢 ∈ 𝐶 → 𝑢 ∈ 𝒫 𝑋)
26	25	ssriv 3985	. . . . . 6 ⊢ 𝐶 ⊆ 𝒫 𝑋
27	26	a1i 11	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ⊆ 𝒫 𝑋)
28	20, 27	ssexd 5323	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ∈ V)
29	28	adantr 479	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → 𝐶 ∈ V)
30		eqid 2730	. . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶
31		eqid 2730	. . . 4 ⊢ ∪ 𝑌 = ∪ 𝑌
32	30, 31	isref 23233	. . 3 ⊢ (𝐶 ∈ V → (𝐶Ref𝑌 ↔ (∪ 𝑌 = ∪ 𝐶 ∧ ∀𝑢 ∈ 𝐶 ∃𝑦 ∈ 𝑌 𝑢 ⊆ 𝑦)))
33	29, 32	syl 17	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → (𝐶Ref𝑌 ↔ (∪ 𝑌 = ∪ 𝐶 ∧ ∀𝑢 ∈ 𝐶 ∃𝑦 ∈ 𝑌 𝑢 ⊆ 𝑦)))
34	4, 19, 33	mpbir2and 709	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → 𝐶Ref𝑌)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  ∀wral 3059  ∃wrex 3068  Vcvv 3472   ⊆ wss 3947  𝒫 cpw 4601  {csn 4627  ∪ cuni 4907   class class class wbr 5147  Refcref 23226

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-ref 23229

This theorem is referenced by:  dispcmp  33137