Theorem resdm2 5955

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 5928	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom ◡◡𝐴)
2		relcnv 5835	. . 3 ⊢ Rel ◡◡𝐴
3		resdm 5770	. . 3 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴)
4	2, 3	ax-mp 5	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴
5		dmcnvcnv 5677	. . 3 ⊢ dom ◡◡𝐴 = dom 𝐴
6	5	reseq2i 5723	. 2 ⊢ (𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom 𝐴)
7	1, 4, 6	3eqtr3ri 2826	1 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴

Syntax hints:   = wceq 1520  ^◡ccnv 5434  dom cdm 5435   ↾ cres 5437  Rel wrel 5440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-br 4957  df-opab 5019  df-xp 5441  df-rel 5442  df-cnv 5443  df-dm 5445  df-rn 5446  df-res 5447

This theorem is referenced by:  resdmres  5956  fimacnvinrn  6696  dfrel5  35085