Theorem resdm2 5121

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 5093	. 2 |- ( `' `' A |` dom `' `' A ) = ( A |` dom `' `' A )
2		relcnv 5008	. . 3 |- Rel `' `' A
3		resdm 4948	. . 3 |- ( Rel `' `' A -> ( `' `' A |` dom `' `' A ) = `' `' A )
4	2, 3	ax-mp 5	. 2 |- ( `' `' A |` dom `' `' A ) = `' `' A
5		dmcnvcnv 4853	. . 3 |- dom `' `' A = dom A
6	5	reseq2i 4906	. 2 |- ( A |` dom `' `' A ) = ( A |` dom A )
7	1, 4, 6	3eqtr3ri 2207	1 |- ( A |` dom A ) = `' `' A

Syntax hints:   = wceq 1353   `'ccnv 4627   dom cdm 4628   |` cres 4630   Rel wrel 4633

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640

This theorem is referenced by:  resdmres  5122