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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 553 | . . . 4 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → (𝑥 ∈ 𝐶 ∧ 𝜃)) |
4 | 3 | ex 449 | . . 3 ⊢ (𝜑 → (𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃))) |
5 | 4 | alrimivv 1843 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃))) |
6 | opabresex0d.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
7 | 6 | elexd 3187 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
8 | opabresex0d.y | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉) | |
9 | 8 | elexd 3187 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ V) |
10 | 7, 9 | opabex3d 7037 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V) |
11 | opabbrex 6593 | . 2 ⊢ ((∀𝑥∀𝑦(𝑥𝑅𝑦 → (𝑥 ∈ 𝐶 ∧ 𝜃)) ∧ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝜃)} ∈ V) → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) | |
12 | 5, 10, 11 | syl2anc 691 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 {cab 2596 Vcvv 3173 class class class wbr 4583 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: opabbrfex0d 40331 opabresexd 40332 |
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