Theorem resdm2 5273

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 5245	. 2 |- ( `' `' A |` dom `' `' A ) = ( A |` dom `' `' A )
2		relcnv 5160	. . 3 |- Rel `' `' A
3		resdm 5097	. . 3 |- ( Rel `' `' A -> ( `' `' A |` dom `' `' A ) = `' `' A )
4	2, 3	ax-mp 5	. 2 |- ( `' `' A |` dom `' `' A ) = `' `' A
5		dmcnvcnv 5001	. . 3 |- dom `' `' A = dom A
6	5	reseq2i 5055	. 2 |- ( A |` dom `' `' A ) = ( A |` dom A )
7	1, 4, 6	3eqtr3ri 2268	1 |- ( A |` dom A ) = `' `' A

Syntax hints:   = wceq 1402   `'ccnv 4768   dom cdm 4769   |` cres 4771   Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781

This theorem is referenced by:  resdmres  5274