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Theorem imai 5941
Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
Assertion
Ref Expression
imai ( I “ 𝐴) = 𝐴

Proof of Theorem imai
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfima3 5931 . 2 ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )}
2 df-br 5066 . . . . . . 7 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
3 vex 3497 . . . . . . . 8 𝑦 ∈ V
43ideq 5722 . . . . . . 7 (𝑥 I 𝑦𝑥 = 𝑦)
52, 4bitr3i 279 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
65anbi1ci 627 . . . . 5 ((𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦𝑥𝐴))
76exbii 1844 . . . 4 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝐴))
8 eleq1w 2895 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98equsexvw 2007 . . . 4 (∃𝑥(𝑥 = 𝑦𝑥𝐴) ↔ 𝑦𝐴)
107, 9bitri 277 . . 3 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝑦𝐴)
1110abbii 2886 . 2 {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦𝑦𝐴}
12 abid2 2957 . 2 {𝑦𝑦𝐴} = 𝐴
131, 11, 123eqtri 2848 1 ( I “ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1533  wex 1776  wcel 2110  {cab 2799  cop 4572   class class class wbr 5065   I cid 5458  cima 5557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567
This theorem is referenced by:  rnresi  5942  cnvresid  6432  ecidsn  8341  mbfid  24235  frege131d  40109  frege110  40319  frege133  40342
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