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 | pm4.24 566 | . . 3 ⊢ (𝑥𝑅𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)) | |
2 | 1 | opabbii 5132 | . 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 43485 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)} ∈ V) |
9 | 2, 8 | eqeltrid 2917 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 Vcvv 3494 class class class wbr 5065 {copab 5127 ⟶wf 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 |
This theorem is referenced by: (None) |
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