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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 5564 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
2 | equcom 2013 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
3 | 2 | opabbii 5205 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
4 | 1, 3 | eqtri 2752 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
5 | 4 | reseq1i 5967 | . 2 ⊢ ( I ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) |
6 | resopab 6024 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
7 | 5, 6 | eqtri 2752 | 1 ⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 {copab 5200 I cid 5563 ↾ cres 5668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-res 5678 |
This theorem is referenced by: mptresid 6040 iresn0n0 6043 pospo 18299 eqg0subg 19111 opsrtoslem1 21925 tgphaus 23942 relexp0eq 42907 |
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