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Mirrors > Home > MPE Home > Th. List > opabex2 | Structured version Visualization version GIF version |
Description: Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.) |
Ref | Expression |
---|---|
opabex2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
opabex2.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
opabex2.3 | ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴) |
opabex2.4 | ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) |
Ref | Expression |
---|---|
opabex2 | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabex2.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | opabex2.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | 1, 2 | xpexd 7770 | . 2 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) |
4 | opabex2.3 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴) | |
5 | opabex2.4 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) | |
6 | 4, 5 | opabssxpd 5736 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
7 | 3, 6 | ssexd 5330 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 Vcvv 3478 {copab 5210 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: tgjustf 28496 legval 28607 wksvOLD 29653 mgcoval 32961 satf00 35359 bj-imdirval2lem 37165 rfovcnvfvd 43997 sprsymrelfvlem 47415 |
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