Theorem cnvtrucl0 41232

Description: The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)

Assertion

Ref	Expression
cnvtrucl0	⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)})

Distinct variable groups:   𝑥,𝑦,𝑉   𝑥,𝑋,𝑦

Proof of Theorem cnvtrucl0

Step	Hyp	Ref	Expression
1		idd 24	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (⊤ → ⊤))
2		idd 24	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ◡𝑥) → (⊤ → ⊤))
3		biidd 261	. 2 ⊢ (𝑥 = 𝑋 → (⊤ ↔ ⊤))
4		ssidd 3944	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑋)
5		elex 3450	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		trud 1549	. 2 ⊢ (𝑋 ∈ 𝑉 → ⊤)
7	1, 2, 3, 4, 5, 6	clcnvlem 41231	1 ⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  {cab 2715   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  ∩ cint 4879  ^◡ccnv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831  df-2nd 7832

This theorem is referenced by: (None)