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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 2156. (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 4467 | . . 3 ⊢ (Rel 𝐴 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = 𝐴 |
4 | opabbi2dv.3 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝜓)) | |
5 | 4 | opabbidv 3823 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
6 | 3, 5 | syl5eqr 2086 | 1 ⊢ (𝜑 → 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 〈cop 3378 {copab 3817 Rel wrel 4350 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 df-rel 4352 |
This theorem is referenced by: (None) |
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