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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 5388 | . 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} | |
2 | df-br 4584 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
3 | vex 3176 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 5196 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | 2, 4 | bitr3i 265 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
6 | 5 | anbi2i 726 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦)) |
7 | ancom 465 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) | |
8 | 6, 7 | bitri 263 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
9 | 8 | exbii 1764 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
10 | eleq1 2676 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
11 | 10 | equsexvw 1919 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴) |
12 | 9, 11 | bitri 263 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ 𝑦 ∈ 𝐴) |
13 | 12 | abbii 2726 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴} |
14 | abid2 2732 | . 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
15 | 1, 13, 14 | 3eqtri 2636 | 1 ⊢ ( I “ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 〈cop 4131 class class class wbr 4583 I cid 4948 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: rnresi 5398 cnvresid 5882 ecidsn 7682 mbfid 23209 frege131d 37075 frege110 37287 frege133 37310 |
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