Theorem rescnvcnv 6172

Description: The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

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

Proof of Theorem rescnvcnv

Step	Hyp	Ref	Expression
1		cnvcnv2 6161	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	reseq1i 5944	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = ((𝐴 ↾ V) ↾ 𝐵)
3		resres 5961	. 2 ⊢ ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵))
4		ssv 3960	. . . 4 ⊢ 𝐵 ⊆ V
5		sseqin2 4177	. . . 4 ⊢ (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵)
6	4, 5	mpbi 230	. . 3 ⊢ (V ∩ 𝐵) = 𝐵
7	6	reseq2i 5945	. 2 ⊢ (𝐴 ↾ (V ∩ 𝐵)) = (𝐴 ↾ 𝐵)
8	2, 3, 7	3eqtri 2764	1 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1542  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  ^◡ccnv 5633   ↾ cres 5636

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 2709  ax-sep 5245  ax-pr 5381

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-cnv 5642  df-res 5646

This theorem is referenced by:  cnvcnvres  6173  imacnvcnv  6174  resdm2  6199  resdmres  6200  coires1  6233  f1oresrab  7084