Theorem cnvcnv2 6158

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 6157	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		df-res 5643	. 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V))
3	1, 2	eqtr4i 2763	1 ⊢ ◡◡𝐴 = (𝐴 ↾ V)

Syntax hints:   = wceq 1542  Vcvv 3430   ∩ cin 3889   × cxp 5629  ^◡ccnv 5630   ↾ cres 5633

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 5232  ax-pr 5376

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5637  df-rel 5638  df-cnv 5639  df-res 5643

This theorem is referenced by:  dfrel3  6163  rnresv  6166  rescnvcnv  6169  cocnvcnv1  6223  cocnvcnv2  6224  strfv2d  17171  resnonrel  44019