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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressn2 | Structured version Visualization version GIF version |
Description: A class ' R ' restricted to the singleton of the class ' A ' is the ordered pair class abstraction of the class ' A ' and the sets in relation ' R ' to ' A ' (and not in relation to the singleton ' { A } ' ). (Contributed by Peter Mazsa, 16-Jun-2024.) |
Ref | Expression |
---|---|
ressn2 | ⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfres2 6034 | . 2 ⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)} | |
2 | velsn 4639 | . . . . 5 ⊢ (𝑎 ∈ {𝐴} ↔ 𝑎 = 𝐴) | |
3 | 2 | anbi1i 623 | . . . 4 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝑎𝑅𝑢)) |
4 | eqbrb 37610 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)) | |
5 | 3, 4 | bitri 275 | . . 3 ⊢ ((𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢) ↔ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)) |
6 | 5 | opabbii 5208 | . 2 ⊢ {⟨𝑎, 𝑢⟩ ∣ (𝑎 ∈ {𝐴} ∧ 𝑎𝑅𝑢)} = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)} |
7 | 1, 6 | eqtri 2754 | 1 ⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 {csn 4623 class class class wbr 5141 {copab 5203 ↾ cres 5671 |
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-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-res 5681 |
This theorem is referenced by: (None) |
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