Theorem cnvcnv2 6190

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 6189	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		df-res 5671	. 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V))
3	1, 2	eqtr4i 2788	1 ⊢ ◡◡𝐴 = (𝐴 ↾ V)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3453   ∩ cin 3901   × cxp 5657  ^◡ccnv 5658   ↾ cres 5661

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-res 5671

This theorem is used by:  cnvcnvss  6191  dfrel3  6196  rnresv  6199  rescnvcnv  6204  cocnvcnv1  6258  cocnvcnv2  6259  strfv2d  17297  resnonrel  44434