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Theorem imai 5618
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 5609 . 2 ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )}
2 df-br 4788 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
3 vex 3354 . . . . . . . . 9 𝑦 ∈ V
43ideq 5412 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
52, 4bitr3i 266 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
65anbi2i 609 . . . . . 6 ((𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥𝐴𝑥 = 𝑦))
7 ancom 448 . . . . . 6 ((𝑥𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝑦𝑥𝐴))
86, 7bitri 264 . . . . 5 ((𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦𝑥𝐴))
98exbii 1924 . . . 4 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝐴))
10 eleq1w 2833 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1110equsexvw 2090 . . . 4 (∃𝑥(𝑥 = 𝑦𝑥𝐴) ↔ 𝑦𝐴)
129, 11bitri 264 . . 3 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝑦𝐴)
1312abbii 2888 . 2 {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦𝑦𝐴}
14 abid2 2894 . 2 {𝑦𝑦𝐴} = 𝐴
151, 13, 143eqtri 2797 1 ( I “ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wex 1852  wcel 2145  {cab 2757  cop 4323   class class class wbr 4787   I cid 5157  cima 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263
This theorem is referenced by:  rnresi  5619  cnvresid  6107  ecidsn  7951  mbfid  23623  frege131d  38582  frege110  38793  frege133  38816
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