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Mirrors > Home > MPE Home > Th. List > opabresid | Structured version Visualization version GIF version |
Description: The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.) |
Ref | Expression |
---|---|
opabresid | ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resopab 5696 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
2 | equcom 2065 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | opabbii 4953 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} |
4 | df-id 5261 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
5 | 3, 4 | eqtr4i 2805 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} = I |
6 | 5 | reseq1i 5638 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = ( I ↾ 𝐴) |
7 | 1, 6 | eqtr3i 2804 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1601 ∈ wcel 2107 {copab 4948 I cid 5260 ↾ cres 5357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-res 5367 |
This theorem is referenced by: mptresid 5712 pospo 17359 opsrtoslem1 19880 tgphaus 22328 relexp0eq 38950 |
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