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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 2259. (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 4678 | . . 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 4002 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
6 | 3, 5 | syl5eqr 2187 | 1 ⊢ (𝜑 → 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∈ wcel 1481 〈cop 3535 {copab 3996 Rel wrel 4552 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 df-xp 4553 df-rel 4554 |
This theorem is referenced by: (None) |
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