Theorem resdm2 6253

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 6226	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom ◡◡𝐴)
2		relcnv 6125	. . 3 ⊢ Rel ◡◡𝐴
3		resdm 6046	. . 3 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴)
4	2, 3	ax-mp 5	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴
5		dmcnvcnv 5947	. . 3 ⊢ dom ◡◡𝐴 = dom 𝐴
6	5	reseq2i 5997	. 2 ⊢ (𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom 𝐴)
7	1, 4, 6	3eqtr3ri 2772	1 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴

Syntax hints:   = wceq 1537  ^◡ccnv 5688  dom cdm 5689   ↾ cres 5691  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701

This theorem is referenced by:  resdmres  6254  fimacnvinrn  7091  dfrel5  38328