Theorem cnvcnvres 6199

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 5996	. . 3 ⊢ Rel (𝐴 ↾ 𝐵)
2		dfrel2 6180	. . 3 ⊢ (Rel (𝐴 ↾ 𝐵) ↔ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	mpbi 233	. 2 ⊢ ◡◡(𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
4		rescnvcnv 6198	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
5	3, 4	eqtr4i 2787	1 ⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5650   ↾ cres 5653  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-res 5663

This theorem is used by: (None)