Theorem resdm2 5111

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 5083	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom ◡◡𝐴)
2		relcnv 4999	. . 3 ⊢ Rel ◡◡𝐴
3		resdm 4939	. . 3 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴)
4	2, 3	ax-mp 5	. 2 ⊢ (◡◡𝐴 ↾ dom ◡◡𝐴) = ◡◡𝐴
5		dmcnvcnv 4844	. . 3 ⊢ dom ◡◡𝐴 = dom 𝐴
6	5	reseq2i 4897	. 2 ⊢ (𝐴 ↾ dom ◡◡𝐴) = (𝐴 ↾ dom 𝐴)
7	1, 4, 6	3eqtr3ri 2205	1 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴

Syntax hints:   = wceq 1353  ^◡ccnv 4619  dom cdm 4620   ↾ cres 4622  Rel wrel 4625

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631  df-res 4632

This theorem is referenced by:  resdmres  5112