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Theorem dissnref 23536
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 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
Assertion
Ref Expression
dissnref ((𝑋𝑉 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.
StepHypRef Expression
1 simpr 484 . . 3 ((𝑋𝑉 𝑌 = 𝑋) → 𝑌 = 𝑋)
2 dissnref.c . . . 4 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
32unisngl 23535 . . 3 𝑋 = 𝐶
41, 3eqtrdi 2793 . 2 ((𝑋𝑉 𝑌 = 𝑋) → 𝑌 = 𝐶)
5 simplr 769 . . . . . 6 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → 𝑢 = {𝑥})
6 simprr 773 . . . . . . 7 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → 𝑥𝑦)
76snssd 4809 . . . . . 6 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → {𝑥} ⊆ 𝑦)
85, 7eqsstrd 4018 . . . . 5 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → 𝑢𝑦)
9 simplr 769 . . . . . . 7 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑥𝑋)
10 simp-4r 784 . . . . . . 7 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑌 = 𝑋)
119, 10eleqtrrd 2844 . . . . . 6 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 𝑌)
12 eluni2 4911 . . . . . 6 (𝑥 𝑌 ↔ ∃𝑦𝑌 𝑥𝑦)
1311, 12sylib 218 . . . . 5 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → ∃𝑦𝑌 𝑥𝑦)
148, 13reximddv 3171 . . . 4 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → ∃𝑦𝑌 𝑢𝑦)
152eqabri 2885 . . . . . 6 (𝑢𝐶 ↔ ∃𝑥𝑋 𝑢 = {𝑥})
1615biimpi 216 . . . . 5 (𝑢𝐶 → ∃𝑥𝑋 𝑢 = {𝑥})
1716adantl 481 . . . 4 (((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) → ∃𝑥𝑋 𝑢 = {𝑥})
1814, 17r19.29a 3162 . . 3 (((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) → ∃𝑦𝑌 𝑢𝑦)
1918ralrimiva 3146 . 2 ((𝑋𝑉 𝑌 = 𝑋) → ∀𝑢𝐶𝑦𝑌 𝑢𝑦)
20 pwexg 5378 . . . . 5 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
21 simpr 484 . . . . . . . . 9 (((𝑢𝐶𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
22 snelpwi 5448 . . . . . . . . . 10 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
2322ad2antlr 727 . . . . . . . . 9 (((𝑢𝐶𝑥𝑋) ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
2421, 23eqeltrd 2841 . . . . . . . 8 (((𝑢𝐶𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
2524, 16r19.29a 3162 . . . . . . 7 (𝑢𝐶𝑢 ∈ 𝒫 𝑋)
2625ssriv 3987 . . . . . 6 𝐶 ⊆ 𝒫 𝑋
2726a1i 11 . . . . 5 (𝑋𝑉𝐶 ⊆ 𝒫 𝑋)
2820, 27ssexd 5324 . . . 4 (𝑋𝑉𝐶 ∈ V)
2928adantr 480 . . 3 ((𝑋𝑉 𝑌 = 𝑋) → 𝐶 ∈ V)
30 eqid 2737 . . . 4 𝐶 = 𝐶
31 eqid 2737 . . . 4 𝑌 = 𝑌
3230, 31isref 23517 . . 3 (𝐶 ∈ V → (𝐶Ref𝑌 ↔ ( 𝑌 = 𝐶 ∧ ∀𝑢𝐶𝑦𝑌 𝑢𝑦)))
3329, 32syl 17 . 2 ((𝑋𝑉 𝑌 = 𝑋) → (𝐶Ref𝑌 ↔ ( 𝑌 = 𝐶 ∧ ∀𝑢𝐶𝑦𝑌 𝑢𝑦)))
344, 19, 33mpbir2and 713 1 ((𝑋𝑉 𝑌 = 𝑋) → 𝐶Ref𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907   class class class wbr 5143  Refcref 23510
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-ref 23513
This theorem is referenced by:  dispcmp  33858
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