Theorem cnvcnv2 6183

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 6182	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		df-res 5664	. 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V))
3	1, 2	eqtr4i 2791	1 ⊢ ◡◡𝐴 = (𝐴 ↾ V)

Syntax hints:   = wceq 1563  Vcvv 3457   ∩ cin 3906   × cxp 5650  ^◡ccnv 5651   ↾ cres 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-res 5664

This theorem is referenced by:  cnvcnvss  6184  dfrel3  6189  rnresv  6192  rescnvcnv  6195  cocnvcnv1  6249  cocnvcnv2  6250  strfv2d  17251  resnonrel  44180