Theorem rescnvcnv 5191

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 5182	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	reseq1i 5001	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = ((𝐴 ↾ V) ↾ 𝐵)
3		resres 5017	. 2 ⊢ ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵))
4		ssv 3246	. . . 4 ⊢ 𝐵 ⊆ V
5		sseqin2 3423	. . . 4 ⊢ (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵)
6	4, 5	mpbi 145	. . 3 ⊢ (V ∩ 𝐵) = 𝐵
7	6	reseq2i 5002	. 2 ⊢ (𝐴 ↾ (V ∩ 𝐵)) = (𝐴 ↾ 𝐵)
8	2, 3, 7	3eqtri 2254	1 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1395  Vcvv 2799   ∩ cin 3196   ⊆ wss 3197  ^◡ccnv 4718   ↾ cres 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-res 4731

This theorem is referenced by:  cnvcnvres  5192  imacnvcnv  5193  resdm2  5219  resdmres  5220  coires1  5246  cocnvres  5253  f1oresrab  5800