Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opabssi | Structured version Visualization version GIF version |
Description: Sufficient condition for a collection of ordered pairs to be a subclass of a relation. (Contributed by Peter Mazsa, 21-Oct-2019.) (Revised by Thierry Arnoux, 18-Feb-2022.) |
Ref | Expression |
---|---|
opabssi.1 | ⊢ (𝜑 → 〈𝑥, 𝑦〉 ∈ 𝐴) |
Ref | Expression |
---|---|
opabssi | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-opab 5102 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | eleq1 2818 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
3 | 2 | biimprd 251 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑧 ∈ 𝐴)) |
4 | opabssi.1 | . . . . 5 ⊢ (𝜑 → 〈𝑥, 𝑦〉 ∈ 𝐴) | |
5 | 3, 4 | impel 509 | . . . 4 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ 𝐴) |
6 | 5 | exlimivv 1940 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ 𝐴) |
7 | 6 | abssi 3969 | . 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ 𝐴 |
8 | 1, 7 | eqsstri 3921 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 {cab 2714 ⊆ wss 3853 〈cop 4533 {copab 5101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-in 3860 df-ss 3870 df-opab 5102 |
This theorem is referenced by: opabid2ss 30627 |
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