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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opabssvv | Structured version Visualization version GIF version |
Description: A variant of relopabiv 5674 (which could be proved from it, similarly to relxp 5553 from xpss 5551). (Contributed by BJ, 28-Dec-2023.) |
Ref | Expression |
---|---|
bj-opabssvv | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3404 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | vex 3404 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | pm3.2i 474 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V)) |
5 | 4 | ssopab2i 5415 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} |
6 | df-xp 5541 | . 2 ⊢ (V × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
7 | 5, 6 | sseqtrri 3924 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2114 Vcvv 3400 ⊆ wss 3853 {copab 5102 × cxp 5533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-v 3402 df-in 3860 df-ss 3870 df-opab 5103 df-xp 5541 |
This theorem is referenced by: (None) |
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