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Theorem cnvtrucl0 38248
 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 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ⊤)} = {𝑦 ∣ (𝑋𝑦 ∧ ⊤)})
Distinct variable groups:   𝑥,𝑦,𝑉   𝑥,𝑋,𝑦

Proof of Theorem cnvtrucl0
StepHypRef Expression
1 idd 24 . 2 ((𝑋𝑉𝑥 = (𝑦 ∪ (𝑋𝑋))) → (⊤ → ⊤))
2 idd 24 . 2 ((𝑋𝑉𝑦 = 𝑥) → (⊤ → ⊤))
3 biidd 252 . 2 (𝑥 = 𝑋 → (⊤ ↔ ⊤))
4 ssid 3657 . . 3 𝑋𝑋
54a1i 11 . 2 (𝑋𝑉𝑋𝑋)
6 elex 3243 . 2 (𝑋𝑉𝑋 ∈ V)
7 a1tru 1540 . 2 (𝑋𝑉 → ⊤)
81, 2, 3, 5, 6, 7clcnvlem 38247 1 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ⊤)} = {𝑦 ∣ (𝑋𝑦 ∧ ⊤)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523  ⊤wtru 1524   ∈ wcel 2030  {cab 2637   ∖ cdif 3604   ∪ cun 3605   ⊆ wss 3607  ∩ cint 4507  ◡ccnv 5142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-1st 7210  df-2nd 7211 This theorem is referenced by: (None)
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