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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 2025 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
3 | 2 | opabbii 5097 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
4 | 1, 3 | eqtri 2821 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
5 | 4 | reseq1i 5814 | . 2 ⊢ ( I ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) |
6 | resopab 5869 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
7 | 5, 6 | eqtri 2821 | 1 ⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 {copab 5092 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 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-res 5531 |
This theorem is referenced by: mptresid 5885 iresn0n0 5890 pospo 17575 opsrtoslem1 20723 tgphaus 22722 relexp0eq 40402 |
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