Theorem resdm2 5227

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 5199	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom ◡◡𝐴)
2		relcnv 5114	. . 3 ⊢ Rel ◡◡𝐴
3		resdm 5052	. . 3 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴)
4	2, 3	ax-mp 5	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴
5		dmcnvcnv 4956	. . 3 ⊢ dom ◡◡𝐴 = dom 𝐴
6	5	reseq2i 5010	. 2 ⊢ (𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom 𝐴)
7	1, 4, 6	3eqtr3ri 2261	1 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴

Syntax hints:   = wceq 1397  ^◡ccnv 4724  dom cdm 4725   ↾ cres 4727  Rel wrel 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737

This theorem is referenced by:  resdmres  5228