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Mirrors > Home > MPE Home > Th. List > opabresid | Structured version Visualization version GIF version |
Description: The restricted identity relation expressed as an ordered-pair class abstraction. (Contributed by FL, 25-Apr-2012.) |
Ref | Expression |
---|---|
opabresid | ⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-id 5425 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
2 | equcom 2029 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
3 | 2 | opabbii 5094 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
4 | 1, 3 | eqtri 2761 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
5 | 4 | reseq1i 5815 | . 2 ⊢ ( I ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) |
6 | resopab 5870 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
7 | 5, 6 | eqtri 2761 | 1 ⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∈ wcel 2113 {copab 5089 I cid 5424 ↾ cres 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-rab 3062 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-res 5531 |
This theorem is referenced by: mptresid 5886 iresn0n0 5891 pospo 17692 opsrtoslem1 20859 tgphaus 22861 relexp0eq 40839 |
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