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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 7454 | . 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 5753 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
7 | 3, 6 | ssexd 5192 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 {copab 5092 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: tgjustf 26267 legval 26378 wksv 27409 mgcoval 30694 satf00 32734 bj-imdirval2lem 34597 rfovcnvfvd 40708 sprsymrelfvlem 44007 |
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