Theorem cnvcnv2 6044

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 6043	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		df-res 5561	. 2 ⊢ (𝐴 ↾ V) = (𝐴 ∩ (V × V))
3	1, 2	eqtr4i 2847	1 ⊢ ◡◡𝐴 = (𝐴 ↾ V)

Syntax hints:   = wceq 1533  Vcvv 3494   ∩ cin 3934   × cxp 5547  ^◡ccnv 5548   ↾ cres 5551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-res 5561

This theorem is referenced by:  dfrel3  6049  rnresv  6052  rescnvcnv  6055  cocnvcnv1  6104  cocnvcnv2  6105  strfv2d  16523  resnonrel  39945