Theorem cnvcnv2 6152

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 6151	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		df-res 5637	. 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V))
3	1, 2	eqtr4i 2763	1 ⊢ ◡◡𝐴 = (𝐴 ↾ V)

Syntax hints:   = wceq 1542  Vcvv 3441   ∩ cin 3901   × cxp 5623  ^◡ccnv 5624   ↾ cres 5627

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-res 5637

This theorem is referenced by:  dfrel3  6157  rnresv  6160  rescnvcnv  6163  cocnvcnv1  6217  cocnvcnv2  6218  strfv2d  17133  resnonrel  43911