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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 2308. (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 4783 | . . 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 4091 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 〈𝑥, 𝑦〉 ∈ 𝐴} = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
6 | 3, 5 | eqtr3id 2236 | 1 ⊢ (𝜑 → 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 〈cop 3617 {copab 4085 Rel wrel 4656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2758 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-opab 4087 df-xp 4657 df-rel 4658 |
This theorem is referenced by: (None) |
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