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Theorem imai 4624
 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 4614 . 2 ( I “ A) = {yx(x A x, y I )}
2 df-br 3756 . . . . . . . 8 (x I y ↔ ⟨x, y I )
3 vex 2554 . . . . . . . . 9 y V
43ideq 4431 . . . . . . . 8 (x I yx = y)
52, 4bitr3i 175 . . . . . . 7 (⟨x, y I ↔ x = y)
65anbi2i 430 . . . . . 6 ((x A x, y I ) ↔ (x A x = y))
7 ancom 253 . . . . . 6 ((x A x = y) ↔ (x = y x A))
86, 7bitri 173 . . . . 5 ((x A x, y I ) ↔ (x = y x A))
98exbii 1493 . . . 4 (x(x A x, y I ) ↔ x(x = y x A))
10 eleq1 2097 . . . . 5 (x = y → (x Ay A))
113, 10ceqsexv 2587 . . . 4 (x(x = y x A) ↔ y A)
129, 11bitri 173 . . 3 (x(x A x, y I ) ↔ y A)
1312abbii 2150 . 2 {yx(x A x, y I )} = {yy A}
14 abid2 2155 . 2 {yy A} = A
151, 13, 143eqtri 2061 1 ( I “ A) = A
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ⟨cop 3370   class class class wbr 3755   I cid 4016   “ cima 4291 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301 This theorem is referenced by:  rnresi  4625  cnvresid  4916  ecidsn  6089
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