Theorem cnvcnvres 5598

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 5426	. . 3 ⊢ Rel (𝐴 ↾ 𝐵)
2		dfrel2 5583	. . 3 ⊢ (Rel (𝐴 ↾ 𝐵) ↔ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	mpbi 220	. 2 ⊢ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
4		rescnvcnv 5597	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
5	3, 4	eqtr4i 2647	1 ⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1483  ^◡ccnv 5113   ↾ cres 5116  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-res 5126

This theorem is referenced by: (None)