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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 5578 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
| 2 | equcom 2017 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
| 3 | 2 | opabbii 5210 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} | 
| 4 | 1, 3 | eqtri 2765 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} | 
| 5 | 4 | reseq1i 5993 | . 2 ⊢ ( I ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) | 
| 6 | resopab 6052 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
| 7 | 5, 6 | eqtri 2765 | 1 ⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {copab 5205 I cid 5577 ↾ cres 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-res 5697 | 
| This theorem is referenced by: mptresid 6069 iresn0n0 6072 pospo 18390 eqg0subg 19214 opsrtoslem1 22079 tgphaus 24125 relexp0eq 43714 | 
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