Theorem bj-cnvcnv2 38192

Description: The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.) Do not depend on cnvcnv 6202 in order to prove it from cnvcnv2 6203. (Revised by BJ, 16-Oct-2026.) Do not depend on cnvcnv 6202; do not use cnvcnv2 6203. (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 6198	. 2 ⊢ ◡◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
2		relpart 38186	. 2 ⊢ (𝐴 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
3	1, 2	eqtr4i 2789	1 ⊢ ◡◡𝐴 = (𝐴 ↾ V)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   class class class wbr 5108  {copab 5173  ^◡ccnv 5657   ↾ cres 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-res 5670

This theorem is used by:  bj-cnvcnv  38193