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Mirrors > Home > MPE Home > Th. List > Mathboxes > opabbrfex0d | Structured version Visualization version GIF version |
Description: A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-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 |
---|---|
opabbrfex0d | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.24 566 | . . 3 ⊢ (𝑥𝑅𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)) | |
2 | 1 | opabbii 5133 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)} |
3 | opabresex0d.x | . . 3 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) | |
4 | opabresex0d.t | . . 3 ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃) | |
5 | opabresex0d.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉) | |
6 | opabresex0d.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
7 | 3, 4, 5, 6 | opabresex0d 43504 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)} ∈ V) |
8 | 2, 7 | eqeltrid 2917 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 {cab 2799 Vcvv 3494 class class class wbr 5066 {copab 5128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 |
This theorem is referenced by: (None) |
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