Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 5120 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | eleq1 2898 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
3 | 2 | biimprd 250 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑧 ∈ 𝐴)) |
4 | opabssi.1 | . . . . 5 ⊢ (𝜑 → 〈𝑥, 𝑦〉 ∈ 𝐴) | |
5 | 3, 4 | impel 508 | . . . 4 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ 𝐴) |
6 | 5 | exlimivv 1927 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ 𝐴) |
7 | 6 | abssi 4044 | . 2 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ 𝐴 |
8 | 1, 7 | eqsstri 3999 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∃wex 1774 ∈ wcel 2108 {cab 2797 ⊆ wss 3934 〈cop 4565 {copab 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-in 3941 df-ss 3950 df-opab 5120 |
This theorem is referenced by: opabid2ss 30357 |
Copyright terms: Public domain | W3C validator |