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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opabresex0d | Structured version Visualization version GIF version |
Description: A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.) |
Ref | Expression |
---|---|
opabresex0d.x | ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) |
opabresex0d.t | ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃) |
opabresex0d.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉) |
opabresex0d.c | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
Ref | Expression |
---|---|
opabresex0d | ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabresex0d.x | . . . . 5 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) | |
2 | opabresex0d.t | . . . . 5 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃) | |
3 | 1, 2 | jca 513 | . . . 4 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ 𝐶 ∧ 𝜃)) |
4 | 3 | ex 414 | . . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃))) |
5 | 4 | alrimivv 1932 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃))) |
6 | opabresex0d.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
7 | opabresex0d.y | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉) | |
8 | 7 | elexd 3495 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ V) |
9 | 6, 8 | opabex3d 7952 | . 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V) |
10 | opabbrex 7460 | . 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃)) ∧ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) | |
11 | 5, 9, 10 | syl2anc 585 | 1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 ∈ wcel 2107 {cab 2710 Vcvv 3475 class class class wbr 5149 {copab 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-opab 5212 df-xp 5683 df-rel 5684 |
This theorem is referenced by: opabbrfex0d 45994 opabresexd 45995 |
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