| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5555 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
| 2 | equcom 2047 | . . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
| 3 | 2 | opabbii 5177 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
| 4 | 1, 3 | eqtri 2785 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} |
| 5 | 4 | reseq1i 5973 | . 2 ⊢ ( I ↾ 𝐴) = ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) |
| 6 | resopab 6035 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
| 7 | 5, 6 | eqtri 2785 | 1 ⊢ ( I ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: ∧ wa 400 = wceq 1569 ∈ wcel 2142 {copab 5172 I cid 5554 ↾ cres 5662 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-pr 5403 |
| This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-clab 2741 df-cleq 2754 df-clel 2837 df-rab 3416 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-opab 5173 df-id 5555 df-xp 5666 df-rel 5667 df-res 5672 |
| This theorem is used by: mptresid 6052 iresn0n0 6055 pospo 18405 eqg0subg 19273 opsrtoslem1 22217 tgphaus 24285 relexp0eq 44455 |
| Copyright terms: Public domain | W3C validator |