Theorem rescnvcnv 5197

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 5188	. . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2	1	reseq1i 5007	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = ((𝐴 ↾ V) ↾ 𝐵)
3		resres 5023	. 2 ⊢ ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵))
4		ssv 3247	. . . 4 ⊢ 𝐵 ⊆ V
5		sseqin2 3424	. . . 4 ⊢ (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵)
6	4, 5	mpbi 145	. . 3 ⊢ (V ∩ 𝐵) = 𝐵
7	6	reseq2i 5008	. 2 ⊢ (𝐴 ↾ (V ∩ 𝐵)) = (𝐴 ↾ 𝐵)
8	2, 3, 7	3eqtri 2254	1 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1395  Vcvv 2800   ∩ cin 3197   ⊆ wss 3198  ^◡ccnv 4722   ↾ cres 4725

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 4205  ax-pow 4262  ax-pr 4297

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 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-res 4735

This theorem is referenced by:  cnvcnvres  5198  imacnvcnv  5199  resdm2  5225  resdmres  5226  coires1  5252  cocnvres  5259  f1oresrab  5808