Theorem cnvcnv2 6180

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 6179	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		df-res 5659	. 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V))
3	1, 2	eqtr4i 2786	1 ⊢ ◡◡𝐴 = (𝐴 ↾ V)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3450   ∩ cin 3897   × cxp 5645  ^◡ccnv 5646   ↾ cres 5649

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 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-res 5659

This theorem is used by:  cnvcnvss  6181  dfrel3  6186  rnresv  6189  rescnvcnv  6194  cocnvcnv1  6248  cocnvcnv2  6249  strfv2d  17341  resnonrel  44536