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Theorem imai 4708
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 4698 . 2 ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )}
2 df-br 3792 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
3 vex 2577 . . . . . . . . 9 𝑦 ∈ V
43ideq 4515 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
52, 4bitr3i 179 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
65anbi2i 438 . . . . . 6 ((𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥𝐴𝑥 = 𝑦))
7 ancom 257 . . . . . 6 ((𝑥𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝑦𝑥𝐴))
86, 7bitri 177 . . . . 5 ((𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦𝑥𝐴))
98exbii 1512 . . . 4 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝐴))
10 eleq1 2116 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
113, 10ceqsexv 2610 . . . 4 (∃𝑥(𝑥 = 𝑦𝑥𝐴) ↔ 𝑦𝐴)
129, 11bitri 177 . . 3 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝑦𝐴)
1312abbii 2169 . 2 {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦𝑦𝐴}
14 abid2 2174 . 2 {𝑦𝑦𝐴} = 𝐴
151, 13, 143eqtri 2080 1 ( I “ 𝐴) = 𝐴
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  cop 3405   class class class wbr 3791   I cid 4052  cima 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by:  rnresi  4709  cnvresid  5000  ecidsn  6183
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