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Theorem dissnref 21141
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 476 . . 3 ((𝑋𝑉 𝑌 = 𝑋) → 𝑌 = 𝑋)
2 dissnref.c . . . 4 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
32unisngl 21140 . . 3 𝑋 = 𝐶
41, 3syl6eq 2660 . 2 ((𝑋𝑉 𝑌 = 𝑋) → 𝑌 = 𝐶)
5 simplr 788 . . . . . 6 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → 𝑢 = {𝑥})
6 simprr 792 . . . . . . 7 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → 𝑥𝑦)
76snssd 4281 . . . . . 6 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → {𝑥} ⊆ 𝑦)
85, 7eqsstrd 3602 . . . . 5 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → 𝑢𝑦)
9 simplr 788 . . . . . . 7 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑥𝑋)
10 simp-4r 803 . . . . . . 7 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑌 = 𝑋)
119, 10eleqtrrd 2691 . . . . . 6 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 𝑌)
12 eluni2 4376 . . . . . 6 (𝑥 𝑌 ↔ ∃𝑦𝑌 𝑥𝑦)
1311, 12sylib 207 . . . . 5 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → ∃𝑦𝑌 𝑥𝑦)
148, 13reximddv 3001 . . . 4 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → ∃𝑦𝑌 𝑢𝑦)
152abeq2i 2722 . . . . . 6 (𝑢𝐶 ↔ ∃𝑥𝑋 𝑢 = {𝑥})
1615biimpi 205 . . . . 5 (𝑢𝐶 → ∃𝑥𝑋 𝑢 = {𝑥})
1716adantl 481 . . . 4 (((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) → ∃𝑥𝑋 𝑢 = {𝑥})
1814, 17r19.29a 3060 . . 3 (((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) → ∃𝑦𝑌 𝑢𝑦)
1918ralrimiva 2949 . 2 ((𝑋𝑉 𝑌 = 𝑋) → ∀𝑢𝐶𝑦𝑌 𝑢𝑦)
20 pwexg 4776 . . . . 5 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
21 simpr 476 . . . . . . . . 9 (((𝑢𝐶𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
22 snelpwi 4839 . . . . . . . . . 10 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
2322ad2antlr 759 . . . . . . . . 9 (((𝑢𝐶𝑥𝑋) ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
2421, 23eqeltrd 2688 . . . . . . . 8 (((𝑢𝐶𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
2524, 16r19.29a 3060 . . . . . . 7 (𝑢𝐶𝑢 ∈ 𝒫 𝑋)
2625ssriv 3572 . . . . . 6 𝐶 ⊆ 𝒫 𝑋
2726a1i 11 . . . . 5 (𝑋𝑉𝐶 ⊆ 𝒫 𝑋)
2820, 27ssexd 4733 . . . 4 (𝑋𝑉𝐶 ∈ V)
2928adantr 480 . . 3 ((𝑋𝑉 𝑌 = 𝑋) → 𝐶 ∈ V)
30 eqid 2610 . . . 4 𝐶 = 𝐶
31 eqid 2610 . . . 4 𝑌 = 𝑌
3230, 31isref 21122 . . 3 (𝐶 ∈ V → (𝐶Ref𝑌 ↔ ( 𝑌 = 𝐶 ∧ ∀𝑢𝐶𝑦𝑌 𝑢𝑦)))
3329, 32syl 17 . 2 ((𝑋𝑉 𝑌 = 𝑋) → (𝐶Ref𝑌 ↔ ( 𝑌 = 𝐶 ∧ ∀𝑢𝐶𝑦𝑌 𝑢𝑦)))
344, 19, 33mpbir2and 959 1 ((𝑋𝑉 𝑌 = 𝑋) → 𝐶Ref𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  wral 2896  wrex 2897  Vcvv 3173  wss 3540  𝒫 cpw 4108  {csn 4125   cuni 4372   class class class wbr 4583  Refcref 21115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-ref 21118
This theorem is referenced by:  dispcmp  29254
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