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Mirrors > Home > ILE Home > Th. List > opabbi2dv | GIF version |
Description: Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2236. (Contributed by NM, 24-Feb-2014.) |
Ref | Expression |
---|---|
opabbi2dv.1 | ⊢ Rel 𝐴 |
opabbi2dv.3 | ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝜓)) |
Ref | Expression |
---|---|
opabbi2dv | ⊢ (𝜑 → 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabbi2dv.1 | . . 3 ⊢ Rel 𝐴 | |
2 | opabid2 4640 | . . 3 ⊢ (Rel 𝐴 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴 |
4 | opabbi2dv.3 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝜓)) | |
5 | 4 | opabbidv 3964 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
6 | 3, 5 | syl5eqr 2164 | 1 ⊢ (𝜑 → 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 〈cop 3500 {copab 3958 Rel wrel 4514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-xp 4515 df-rel 4516 |
This theorem is referenced by: (None) |
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