Theorem cnvcnvres 6196

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 5995	. . 3 ⊢ Rel (𝐴 ↾ 𝐵)
2		dfrel2 6179	. . 3 ⊢ (Rel (𝐴 ↾ 𝐵) ↔ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	mpbi 233	. 2 ⊢ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
4		rescnvcnv 6195	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
5	3, 4	eqtr4i 2791	1 ⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1563  ^◡ccnv 5651   ↾ cres 5654  Rel wrel 5657

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-ral 3080  df-rex 3090  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: (None)