Theorem cnvcnv2 6191

Description: The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)

Assertion

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

Proof of Theorem cnvcnv2

Step	Hyp	Ref	Expression
1		cnvcnv 6190	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		df-res 5673	. 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V))
3	1, 2	eqtr4i 2787	1 ⊢ ◡◡𝐴 = (𝐴 ↾ V)

Syntax hints:   = wceq 1568  Vcvv 3453   ∩ cin 3903   × cxp 5659  ^◡ccnv 5660   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-cnv 5669  df-res 5673

This theorem is referenced by:  cnvcnvss  6192  dfrel3  6197  rnresv  6200  rescnvcnv  6205  cocnvcnv1  6259  cocnvcnv2  6260  strfv2d  17260  resnonrel  44288