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Mirrors > Home > MPE Home > Th. List > imai | Structured version Visualization version GIF version |
Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.) |
Ref | Expression |
---|---|
imai | ⊢ ( I “ 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfima3 6060 | . 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} | |
2 | df-br 5148 | . . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I ) | |
3 | vex 3479 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 5850 | . . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | 2, 4 | bitr3i 277 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦) |
6 | 5 | anbi1ci 627 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
7 | 6 | exbii 1851 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
8 | eleq1w 2817 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
9 | 8 | equsexvw 2009 | . . . 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 2803 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴} |
12 | abid2 2872 | . 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
13 | 1, 11, 12 | 3eqtri 2765 | 1 ⊢ ( I “ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ⟨cop 4633 class class class wbr 5147 I cid 5572 “ cima 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 |
This theorem is referenced by: rnresi 6071 cnvresid 6624 ecidsn 8752 eqg0subgecsn 19068 mbfid 25134 frege131d 42448 frege110 42657 frege133 42680 |
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