Theorem rescnvcnv 6205

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 6191	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	reseq1i 5974	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = ((𝐴 ↾ V) ↾ 𝐵)
3		resres 5991	. 2 ⊢ ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵))
4		ssv 3961	. . . 4 ⊢ 𝐵 ⊆ V
5		sseqin2 4176	. . . 4 ⊢ (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵)
6	4, 5	mpbi 233	. . 3 ⊢ (V ∩ 𝐵) = 𝐵
7	6	reseq2i 5975	. 2 ⊢ (𝐴 ↾ (V ∩ 𝐵)) = (𝐴 ↾ 𝐵)
8	2, 3, 7	3eqtri 2790	1 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ^◡ccnv 5660   ↾ cres 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-res 5673

This theorem is used by:  cnvcnvres  6206  imacnvcnv  6207  resdm2  6232  resdmres  6233  coires1  6266  f1oresrab  7123