Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 | pm4.24 675 | . . 3 ⊢ (𝑥𝑅𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)) | |
2 | 1 | opabbii 4717 | . 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 41304 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝑥𝑅𝑦)} ∈ V) |
8 | 2, 7 | syl5eqel 2705 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 {cab 2608 Vcvv 3200 class class class wbr 4653 {copab 4712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |