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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opabbrfexd | Structured version Visualization version GIF version |
Description: A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.) |
Ref | Expression |
---|---|
opabresexd.x | ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) |
opabresexd.y | ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵) |
opabresexd.a | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈) |
opabresexd.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉) |
opabresexd.c | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
Ref | Expression |
---|---|
opabbrfexd | ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.24 562 | . . 3 ⊢ (𝑥𝑅𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)) | |
2 | 1 | opabbii 5210 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)} |
3 | opabresexd.x | . . 3 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) | |
4 | opabresexd.y | . . 3 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵) | |
5 | opabresexd.a | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈) | |
6 | opabresexd.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉) | |
7 | opabresexd.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
8 | 3, 4, 5, 6, 7 | opabresexd 46729 | . 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)} ∈ V) |
9 | 2, 8 | eqeltrid 2829 | 1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3463 class class class wbr 5143 {copab 5205 ⟶wf 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-fun 6544 df-fn 6545 df-f 6546 |
This theorem is referenced by: (None) |
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