Theorem bj-cnvcnv2 38063

Description: The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.) Do not depend on cnvcnv 6185 in order to prove it from cnvcnv2 6186. (Revised by BJ, 16-Oct-2026.) Do not depend on cnvcnv 6185. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cnvcnv2	⊢ ◡◡𝐴 = (𝐴 ↾ V)

Proof of Theorem bj-cnvcnv2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvcnv3 6181	. 2 ⊢ ◡◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
2		relpart 38058	. 2 ⊢ (𝐴 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
3	1, 2	eqtr4i 2786	1 ⊢ ◡◡𝐴 = (𝐴 ↾ V)

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3450   class class class wbr 5102  {copab 5166  ^◡ccnv 5649   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-res 5662

This theorem is used by:  bj-cnvcnv  38064