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Theorem cnvtrucl0 37750
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 3616 . . 3 𝑋𝑋
54a1i 11 . 2 (𝑋𝑉𝑋𝑋)
6 elex 3207 . 2 (𝑋𝑉𝑋 ∈ V)
7 a1tru 1498 . 2 (𝑋𝑉 → ⊤)
81, 2, 3, 5, 6, 7clcnvlem 37749 1 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ⊤)} = {𝑦 ∣ (𝑋𝑦 ∧ ⊤)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wtru 1482  wcel 1988  {cab 2606  cdif 3564  cun 3565  wss 3567   cint 4466  ccnv 5103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-1st 7153  df-2nd 7154
This theorem is referenced by: (None)
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