Theorem cnvcnvres 6163

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 5964	. . 3 ⊢ Rel (𝐴 ↾ 𝐵)
2		dfrel2 6147	. . 3 ⊢ (Rel (𝐴 ↾ 𝐵) ↔ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	mpbi 231	. 2 ⊢ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
4		rescnvcnv 6162	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
5	3, 4	eqtr4i 2766	1 ⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1547  ^◡ccnv 5624   ↾ cres 5627  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-res 5637

This theorem is referenced by: (None)