Theorem resdm2 3496

Description: A class restricted to its domain equals its double converse.

Assertion

Ref	Expression
resdm2	|- (A |` dom A) = `'`'A

Proof of Theorem resdm2

Step	Hyp	Ref	Expression
1		rescnvcnv 3493	. 2 |- (`'`'A |` dom `'`'A) = (A |` dom `'`'A)
2		relcnv 3435	. . 3 |- Rel `'`'A
3		resdm 3393	. . 3 |- (Rel `'`'A -> (`'`'A |` dom `'`'A) = `'`'A)
4	2, 3	ax-mp 7	. 2 |- (`'`'A |` dom `'`'A) = `'`'A
5		dmcnvcnv 3336	. . 3 |- dom `'`'A = dom A
6		reseq2 3369	. . 3 |- (dom `'`'A = dom A -> (A |` dom `'`'A) = (A |` dom A))
7	5, 6	ax-mp 7	. 2 |- (A |` dom `'`'A) = (A |` dom A)
8	1, 4, 7	3eqtr3r 1504	1 |- (A |` dom A) = `'`'A

Syntax hints:   = wceq 956  `'ccnv 3169  dom cdm 3170   |` cres 3172  Rel wrel 3175

This theorem is referenced by:  resdmres 3497

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190

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