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Theorem imai 5720
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 5711 . 2 ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )}
2 df-br 4875 . . . . . . . 8 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
3 vex 3418 . . . . . . . . 9 𝑦 ∈ V
43ideq 5508 . . . . . . . 8 (𝑥 I 𝑦𝑥 = 𝑦)
52, 4bitr3i 269 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
65anbi2i 618 . . . . . 6 ((𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥𝐴𝑥 = 𝑦))
7 ancom 454 . . . . . 6 ((𝑥𝐴𝑥 = 𝑦) ↔ (𝑥 = 𝑦𝑥𝐴))
86, 7bitri 267 . . . . 5 ((𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦𝑥𝐴))
98exbii 1949 . . . 4 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦𝑥𝐴))
10 eleq1w 2890 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1110equsexvw 2111 . . . 4 (∃𝑥(𝑥 = 𝑦𝑥𝐴) ↔ 𝑦𝐴)
129, 11bitri 267 . . 3 (∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝑦𝐴)
1312abbii 2945 . 2 {𝑦 ∣ ∃𝑥(𝑥𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦𝑦𝐴}
14 abid2 2951 . 2 {𝑦𝑦𝐴} = 𝐴
151, 13, 143eqtri 2854 1 ( I “ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1658  wex 1880  wcel 2166  {cab 2812  cop 4404   class class class wbr 4874   I cid 5250  cima 5346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356
This theorem is referenced by:  rnresi  5721  cnvresid  6202  ecidsn  8061  mbfid  23802  frege131d  38898  frege110  39108  frege133  39131
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