Theorem cnvtrucl0 44205

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 264	. 2 ⊢ (𝑥 = 𝑋 → (⊤ ↔ ⊤))
4		ssidd 3961	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑋)
5		elex 3477	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
6		trud 1572	. 2 ⊢ (𝑋 ∈ 𝑉 → ⊤)
7	1, 2, 3, 4, 5, 6	clcnvlem 44204	1 ⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562  ⊤wtru 1563   ∈ wcel 2144  {cab 2742   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ∩ cint 4907  ^◡ccnv 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-iota 6479  df-fun 6525  df-fv 6531  df-1st 7972  df-2nd 7973

This theorem is referenced by: (None)