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Mirrors > Home > MPE Home > Th. List > opabresidOLD | Structured version Visualization version GIF version |
Description: Obsolete version of opabresid 5910 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 5895 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} | |
2 | equcom 2024 | . . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | opabbii 5126 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} |
4 | df-id 5453 | . . . 4 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
5 | 3, 4 | eqtr4i 2846 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} = I |
6 | 5 | reseq1i 5842 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 = 𝑥} ↾ 𝐴) = ( I ↾ 𝐴) |
7 | 1, 6 | eqtr3i 2845 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥)} = ( I ↾ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 {copab 5121 I cid 5452 ↾ cres 5550 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-res 5560 |
This theorem is referenced by: mptresidOLD 5913 |
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