Theorem imai 6023

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.

Step	Hyp	Ref	Expression
1		dfima3 6012	. 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )}
2		df-br 5092	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
3		vex 3440	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	3	ideq 5792	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5	2, 4	bitr3i 277	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
6	5	anbi1ci 626	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
7	6	exbii 1849	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
8		eleq1w 2814	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
9	8	equsexvw 2006	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴)
10	7, 9	bitri 275	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝑦 ∈ 𝐴)
11	10	abbii 2798	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴}
12		abid2 2868	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
13	1, 11, 12	3eqtri 2758	1 ⊢ ( I “ 𝐴) = 𝐴

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ⟨cop 4582   class class class wbr 5091   I cid 5510   “ cima 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629

This theorem is referenced by:  rnresi  6024  cnvresid  6560  ecidsn  8680  eqg0subgecsn  19110  mbfid  25564  frege131d  43803  frege110  44012  frege133  44035