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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 7463 | . 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 5782 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (𝐴 × 𝐵)) |
7 | 3, 6 | ssexd 5219 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 Vcvv 3492 {copab 5119 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-opab 5120 df-xp 5554 df-rel 5555 |
This theorem is referenced by: tgjustf 26186 legval 26297 wksv 27328 satf00 32518 bj-imdirval2 34365 rfovcnvfvd 40231 sprsymrelfvlem 43529 |
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