Theorem cnvcnvres 6162

Description: The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)

Assertion

Ref	Expression
cnvcnvres	⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)

Proof of Theorem cnvcnvres

Step	Hyp	Ref	Expression
1		relres 5963	. . 3 ⊢ Rel (𝐴 ↾ 𝐵)
2		dfrel2 6146	. . 3 ⊢ (Rel (𝐴 ↾ 𝐵) ↔ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	mpbi 230	. 2 ⊢ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
4		rescnvcnv 6161	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
5	3, 4	eqtr4i 2761	1 ⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1542  ^◡ccnv 5622   ↾ cres 5625  Rel wrel 5628

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 2707  ax-sep 5240  ax-nul 5250  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 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5629  df-rel 5630  df-cnv 5631  df-res 5635

This theorem is referenced by: (None)