Theorem dmresi 3395

Description: The domain of a restricted identity function.

Assertion

Ref	Expression
dmresi	|- dom ( I |` A) = A

Proof of Theorem dmresi

Step	Hyp	Ref	Expression
1		ssv 2078	. . 3 |- A (_ V
2		dmi 3322	. . 3 |- dom I = V
3	1, 2	sseqtr4 2091	. 2 |- A (_ dom I
4		ssdmres 3377	. 2 |- (A (_ dom I <-> dom ( I |` A) = A)
5	3, 4	mpbi 189	1 |- dom ( I |` A) = A

Syntax hints:   = wceq 955  Vcvv 1808   (_ wss 2044  Icid 2827  dom cdm 3166   |` cres 3168

This theorem is referenced by:  fnresi 3599

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-dm 3184  df-res 3186

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