Theorem resdm2 5160

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 5132	. 2 |- ( `' `' A |` dom `' `' A ) = ( A |` dom `' `' A )
2		relcnv 5047	. . 3 |- Rel `' `' A
3		resdm 4985	. . 3 |- ( Rel `' `' A -> ( `' `' A |` dom `' `' A ) = `' `' A )
4	2, 3	ax-mp 5	. 2 |- ( `' `' A |` dom `' `' A ) = `' `' A
5		dmcnvcnv 4890	. . 3 |- dom `' `' A = dom A
6	5	reseq2i 4943	. 2 |- ( A |` dom `' `' A ) = ( A |` dom A )
7	1, 4, 6	3eqtr3ri 2226	1 |- ( A |` dom A ) = `' `' A

Syntax hints:   = wceq 1364   `'ccnv 4662   dom cdm 4663   |` cres 4665   Rel wrel 4668

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675

This theorem is referenced by:  resdmres  5161