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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 5934 | . 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} | |
2 | df-br 5069 | . . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
3 | vex 3499 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 5725 | . . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | 2, 4 | bitr3i 279 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
6 | 5 | anbi1ci 627 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
7 | 6 | exbii 1848 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
8 | eleq1w 2897 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
9 | 8 | equsexvw 2011 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴) |
10 | 7, 9 | bitri 277 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ 𝑦 ∈ 𝐴) |
11 | 10 | abbii 2888 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴} |
12 | abid2 2959 | . 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
13 | 1, 11, 12 | 3eqtri 2850 | 1 ⊢ ( I “ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 〈cop 4575 class class class wbr 5068 I cid 5461 “ cima 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 |
This theorem is referenced by: rnresi 5945 cnvresid 6435 ecidsn 8344 mbfid 24238 frege131d 40116 frege110 40326 frege133 40349 |
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