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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 5583 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
2 | equcom 2015 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
3 | 2 | opabbii 5215 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
4 | 1, 3 | eqtri 2763 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
5 | 4 | reseq1i 5996 | . 2 ⊢ ( I ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) |
6 | resopab 6054 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
7 | 5, 6 | eqtri 2763 | 1 ⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 {copab 5210 I cid 5582 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-res 5701 |
This theorem is referenced by: mptresid 6071 iresn0n0 6074 pospo 18403 eqg0subg 19227 opsrtoslem1 22097 tgphaus 24141 relexp0eq 43691 |
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