Theorem resdmres 3497

Description: Restriction to the domain of a restriction.

Assertion

Ref	Expression
resdmres	|- (A |` dom ( A |` B)) = (A |` B)

Proof of Theorem resdmres

Step	Hyp	Ref	Expression
1		in12 2224	. . . 4 |- (A i^i ((B X. V) i^i (dom A X. V))) = ((B X. V) i^i (A i^i (dom A X. V)))
2		df-res 3190	. . . . . 6 |- (A |` dom A) = (A i^i (dom A X. V))
3		resdm2 3496	. . . . . 6 |- (A |` dom A) = `'`'A
4	2, 3	eqtr3 1497	. . . . 5 |- (A i^i (dom A X. V)) = `'`'A
5	4	ineq2i 2214	. . . 4 |- ((B X. V) i^i (A i^i (dom A X. V))) = ((B X. V) i^i `'`'A)
6		incom 2208	. . . 4 |- ((B X. V) i^i `'`'A) = (`'`'A i^i (B X. V))
7	1, 5, 6	3eqtr 1499	. . 3 |- (A i^i ((B X. V) i^i (dom A X. V))) = (`'`'A i^i (B X. V))
8		df-res 3190	. . . 4 |- (A |` dom ( A |` B)) = (A i^i (dom ( A |` B) X. V))
9		dmres 3380	. . . . . . 7 |- dom ( A |` B) = (B i^i dom A)
10		xpeq1 3200	. . . . . . 7 |- (dom ( A |` B) = (B i^i dom A) -> (dom ( A |` B) X. V) = ((B i^i dom A) X. V))
11	9, 10	ax-mp 7	. . . . . 6 |- (dom ( A |` B) X. V) = ((B i^i dom A) X. V)
12		xpindir 3271	. . . . . 6 |- ((B i^i dom A) X. V) = ((B X. V) i^i (dom A X. V))
13	11, 12	eqtr 1495	. . . . 5 |- (dom ( A |` B) X. V) = ((B X. V) i^i (dom A X. V))
14	13	ineq2i 2214	. . . 4 |- (A i^i (dom ( A |` B) X. V)) = (A i^i ((B X. V) i^i (dom A X. V)))
15	8, 14	eqtr 1495	. . 3 |- (A |` dom ( A |` B)) = (A i^i ((B X. V) i^i (dom A X. V)))
16		df-res 3190	. . 3 |- (`'`'A |` B) = (`'`'A i^i (B X. V))
17	7, 15, 16	3eqtr4 1505	. 2 |- (A |` dom ( A |` B)) = (`'`'A |` B)
18		rescnvcnv 3493	. 2 |- (`'`'A |` B) = (A |` B)
19	17, 18	eqtr 1495	1 |- (A |` dom ( A |` B)) = (A |` B)

Syntax hints:   = wceq 956  Vcvv 1811   i^i cin 2046   X. cxp 3168  `'ccnv 3169  dom cdm 3170   |` cres 3172

This theorem is referenced by:  imadmres 3498  metres 7823

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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