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Theorem cnvtrucl0 40732
 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 265 . 2 (𝑥 = 𝑋 → (⊤ ↔ ⊤))
4 ssidd 3917 . 2 (𝑋𝑉𝑋𝑋)
5 elex 3428 . 2 (𝑋𝑉𝑋 ∈ V)
6 trud 1548 . 2 (𝑋𝑉 → ⊤)
71, 2, 3, 4, 5, 6clcnvlem 40731 1 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ⊤)} = {𝑦 ∣ (𝑋𝑦 ∧ ⊤)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  {cab 2735   ∖ cdif 3857   ∪ cun 3858   ⊆ wss 3860  ∩ cint 4841  ◡ccnv 5527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6299  df-fun 6342  df-fv 6348  df-1st 7699  df-2nd 7700 This theorem is referenced by: (None)
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