Theorem resdm2 5094

Description: A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
resdm2	⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴

Proof of Theorem resdm2

Step	Hyp	Ref	Expression
1		rescnvcnv 5066	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom ◡◡𝐴)
2		relcnv 4982	. . 3 ⊢ Rel ◡◡𝐴
3		resdm 4923	. . 3 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴)
4	2, 3	ax-mp 5	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴
5		dmcnvcnv 4828	. . 3 ⊢ dom ◡◡𝐴 = dom 𝐴
6	5	reseq2i 4881	. 2 ⊢ (𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom 𝐴)
7	1, 4, 6	3eqtr3ri 2195	1 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴

Syntax hints:   = wceq 1343  ^◡ccnv 4603  dom cdm 4604   ↾ cres 4606  Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616

This theorem is referenced by:  resdmres  5095