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Theorem dissnref 21533
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 479 . . 3 ((𝑋𝑉 𝑌 = 𝑋) → 𝑌 = 𝑋)
2 dissnref.c . . . 4 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
32unisngl 21532 . . 3 𝑋 = 𝐶
41, 3syl6eq 2810 . 2 ((𝑋𝑉 𝑌 = 𝑋) → 𝑌 = 𝐶)
5 simplr 809 . . . . . 6 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → 𝑢 = {𝑥})
6 simprr 813 . . . . . . 7 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → 𝑥𝑦)
76snssd 4485 . . . . . 6 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → {𝑥} ⊆ 𝑦)
85, 7eqsstrd 3780 . . . . 5 ((((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) ∧ (𝑦𝑌𝑥𝑦)) → 𝑢𝑦)
9 simplr 809 . . . . . . 7 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑥𝑋)
10 simp-4r 827 . . . . . . 7 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑌 = 𝑋)
119, 10eleqtrrd 2842 . . . . . 6 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑥 𝑌)
12 eluni2 4592 . . . . . 6 (𝑥 𝑌 ↔ ∃𝑦𝑌 𝑥𝑦)
1311, 12sylib 208 . . . . 5 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → ∃𝑦𝑌 𝑥𝑦)
148, 13reximddv 3156 . . . 4 (((((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) ∧ 𝑥𝑋) ∧ 𝑢 = {𝑥}) → ∃𝑦𝑌 𝑢𝑦)
152abeq2i 2873 . . . . . 6 (𝑢𝐶 ↔ ∃𝑥𝑋 𝑢 = {𝑥})
1615biimpi 206 . . . . 5 (𝑢𝐶 → ∃𝑥𝑋 𝑢 = {𝑥})
1716adantl 473 . . . 4 (((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) → ∃𝑥𝑋 𝑢 = {𝑥})
1814, 17r19.29a 3216 . . 3 (((𝑋𝑉 𝑌 = 𝑋) ∧ 𝑢𝐶) → ∃𝑦𝑌 𝑢𝑦)
1918ralrimiva 3104 . 2 ((𝑋𝑉 𝑌 = 𝑋) → ∀𝑢𝐶𝑦𝑌 𝑢𝑦)
20 pwexg 4999 . . . . 5 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
21 simpr 479 . . . . . . . . 9 (((𝑢𝐶𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
22 snelpwi 5061 . . . . . . . . . 10 (𝑥𝑋 → {𝑥} ∈ 𝒫 𝑋)
2322ad2antlr 765 . . . . . . . . 9 (((𝑢𝐶𝑥𝑋) ∧ 𝑢 = {𝑥}) → {𝑥} ∈ 𝒫 𝑋)
2421, 23eqeltrd 2839 . . . . . . . 8 (((𝑢𝐶𝑥𝑋) ∧ 𝑢 = {𝑥}) → 𝑢 ∈ 𝒫 𝑋)
2524, 16r19.29a 3216 . . . . . . 7 (𝑢𝐶𝑢 ∈ 𝒫 𝑋)
2625ssriv 3748 . . . . . 6 𝐶 ⊆ 𝒫 𝑋
2726a1i 11 . . . . 5 (𝑋𝑉𝐶 ⊆ 𝒫 𝑋)
2820, 27ssexd 4957 . . . 4 (𝑋𝑉𝐶 ∈ V)
2928adantr 472 . . 3 ((𝑋𝑉 𝑌 = 𝑋) → 𝐶 ∈ V)
30 eqid 2760 . . . 4 𝐶 = 𝐶
31 eqid 2760 . . . 4 𝑌 = 𝑌
3230, 31isref 21514 . . 3 (𝐶 ∈ V → (𝐶Ref𝑌 ↔ ( 𝑌 = 𝐶 ∧ ∀𝑢𝐶𝑦𝑌 𝑢𝑦)))
3329, 32syl 17 . 2 ((𝑋𝑉 𝑌 = 𝑋) → (𝐶Ref𝑌 ↔ ( 𝑌 = 𝐶 ∧ ∀𝑢𝐶𝑦𝑌 𝑢𝑦)))
344, 19, 33mpbir2and 995 1 ((𝑋𝑉 𝑌 = 𝑋) → 𝐶Ref𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  Vcvv 3340  wss 3715  𝒫 cpw 4302  {csn 4321   cuni 4588   class class class wbr 4804  Refcref 21507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-ref 21510
This theorem is referenced by:  dispcmp  30235
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