Theorem imai 6078

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 6067	. 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )}
2		df-br 5111	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
3		vex 3459	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	3	ideq 5840	. . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
5	2, 4	bitr3i 280	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
6	5	anbi1ci 637	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
7	6	exbii 1878	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴))
8		eleq1w 2846	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
9	8	equsexvw 2035	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴)
10	7, 9	bitri 278	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ 𝑦 ∈ 𝐴)
11	10	abbii 2830	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴}
12		abid2 2900	. 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴
13	1, 11, 12	3eqtri 2790	1 ⊢ ( I “ 𝐴) = 𝐴

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ⟨cop 4596   class class class wbr 5110   I cid 5557   “ cima 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is referenced by:  rnresi  6079  cnvresid  6617  ecidsn  8754  eqg0subgecsn  19269  mbfid  25775  frege131d  44473  frege110  44682  frege133  44705