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Theorem imai 5383
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 5374 . 2 ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )}
2 df-br 4578 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
3 vex 3175 . . . . . . . . 9 𝑦 ∈ V
43ideq 5183 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
52, 4bitr3i 264 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
65anbi2i 725 . . . . . 6 ((𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥𝐴𝑥 = 𝑦))
7 ancom 464 . . . . . 6 ((𝑥𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝑦𝑥𝐴))
86, 7bitri 262 . . . . 5 ((𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦𝑥𝐴))
98exbii 1763 . . . 4 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝐴))
10 eleq1 2675 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1110equsexvw 1918 . . . 4 (∃𝑥(𝑥 = 𝑦𝑥𝐴) ↔ 𝑦𝐴)
129, 11bitri 262 . . 3 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝑦𝐴)
1312abbii 2725 . 2 {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦𝑦𝐴}
14 abid2 2731 . 2 {𝑦𝑦𝐴} = 𝐴
151, 13, 143eqtri 2635 1 ( I “ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wex 1694  wcel 1976  {cab 2595  cop 4130   class class class wbr 4577   I cid 4937  cima 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040
This theorem is referenced by:  rnresi  5384  cnvresid  5867  ecidsn  7659  mbfid  23153  frege131d  36858  frege110  37070  frege133  37093
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