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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 5469 | . 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} | |
2 | df-br 4654 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
3 | vex 3203 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 5274 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | 2, 4 | bitr3i 266 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
6 | 5 | anbi2i 730 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦)) |
7 | ancom 466 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) | |
8 | 6, 7 | bitri 264 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
9 | 8 | exbii 1774 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
10 | eleq1 2689 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
11 | 10 | equsexvw 1932 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴) |
12 | 9, 11 | bitri 264 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ 𝑦 ∈ 𝐴) |
13 | 12 | abbii 2739 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴} |
14 | abid2 2745 | . 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
15 | 1, 13, 14 | 3eqtri 2648 | 1 ⊢ ( I “ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 {cab 2608 〈cop 4183 class class class wbr 4653 I cid 5023 “ cima 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 |
This theorem is referenced by: rnresi 5479 cnvresid 5968 ecidsn 7795 mbfid 23403 frege131d 38056 frege110 38267 frege133 38290 |
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