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Mirrors > Home > MPE Home > Th. List > opabresidOLD | Structured version Visualization version GIF version |
Description: Obsolete version of opabresid 5917 as of 26-Dec-2023. (Contributed by FL, 25-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
opabresidOLD | ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resopab 5902 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
2 | equcom 2025 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | opabbii 5133 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} |
4 | df-id 5460 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
5 | 3, 4 | eqtr4i 2847 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} = I |
6 | 5 | reseq1i 5849 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = ( I ↾ 𝐴) |
7 | 1, 6 | eqtr3i 2846 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 {copab 5128 I cid 5459 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-res 5567 |
This theorem is referenced by: mptresidOLD 5920 |
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