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Mirrors > Home > ILE Home > Th. List > imai | 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 4879 | . 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} | |
2 | df-br 3925 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
3 | vex 2684 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 4686 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | 2, 4 | bitr3i 185 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
6 | 5 | anbi2i 452 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦)) |
7 | ancom 264 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) | |
8 | 6, 7 | bitri 183 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
9 | 8 | exbii 1584 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
10 | eleq1 2200 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
11 | 3, 10 | ceqsexv 2720 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴) |
12 | 9, 11 | bitri 183 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ 𝑦 ∈ 𝐴) |
13 | 12 | abbii 2253 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴} |
14 | abid2 2258 | . 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
15 | 1, 13, 14 | 3eqtri 2162 | 1 ⊢ ( I “ 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {cab 2123 〈cop 3525 class class class wbr 3924 I cid 4205 “ cima 4537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 |
This theorem is referenced by: rnresi 4891 cnvresid 5192 ecidsn 6469 |
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