Theorem dissnref 22587

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 484	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → ∪ 𝑌 = 𝑋)
2		dissnref.c	. . . 4 ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}
3	2	unisngl 22586	. . 3 ⊢ 𝑋 = ∪ 𝐶
4	1, 3	eqtrdi 2795	. 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 4739	. . . . . 6 ⊢ ((((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑦)) → {𝑥} ⊆ 𝑦)
8	5, 7	eqsstrd 3955	. . . . 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 2842	. . . . . 6 ⊢ (((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 ∈ ∪ 𝑌)
12		eluni2 4840	. . . . . 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 3203	. . . 4 ⊢ (((((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → ∃𝑦 ∈ 𝑌 𝑢 ⊆ 𝑦)
15	2	abeq2i 2874	. . . . . 6 ⊢ (𝑢 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
16	15	biimpi 215	. . . . 5 ⊢ (𝑢 ∈ 𝐶 → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
17	16	adantl 481	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) → ∃𝑥 ∈ 𝑋 𝑢 = {𝑥})
18	14, 17	r19.29a 3217	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) ∧ 𝑢 ∈ 𝐶) → ∃𝑦 ∈ 𝑌 𝑢 ⊆ 𝑦)
19	18	ralrimiva 3107	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → ∀𝑢 ∈ 𝐶 ∃𝑦 ∈ 𝑌 𝑢 ⊆ 𝑦)
20		pwexg 5296	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
21		simpr 484	. . . . . . . . 9 ⊢ (((𝑢 ∈ 𝐶 ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
22		snelpwi 5354	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ∈ 𝒫 𝑋)
23	22	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝑢 ∈ 𝐶 ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
24	21, 23	eqeltrd 2839	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐶 ∧ 𝑥 ∈ 𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
25	24, 16	r19.29a 3217	. . . . . . 7 ⊢ (𝑢 ∈ 𝐶 → 𝑢 ∈ 𝒫 𝑋)
26	25	ssriv 3921	. . . . . 6 ⊢ 𝐶 ⊆ 𝒫 𝑋
27	26	a1i 11	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ⊆ 𝒫 𝑋)
28	20, 27	ssexd 5243	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝐶 ∈ V)
29	28	adantr 480	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∪ 𝑌 = 𝑋) → 𝐶 ∈ V)
30		eqid 2738	. . . 4 ⊢ ∪ 𝐶 = ∪ 𝐶
31		eqid 2738	. . . 4 ⊢ ∪ 𝑌 = ∪ 𝑌
32	30, 31	isref 22568	. . 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 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   class class class wbr 5070  Refcref 22561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-ref 22564

This theorem is referenced by:  dispcmp  31711