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Mirrors > Home > ILE Home > Th. List > opelrn | GIF version |
Description: Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.) |
Ref | Expression |
---|---|
brelrn.1 | ⊢ 𝐴 ∈ V |
brelrn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opelrn | ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ran 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3765 | . 2 ⊢ (𝐴𝐶𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶) | |
2 | brelrn.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | brelrn.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | brelrn 4567 | . 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) |
5 | 1, 4 | sylbir 125 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 Vcvv 2557 〈cop 3378 class class class wbr 3764 ran crn 4346 |
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-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-cnv 4353 df-dm 4355 df-rn 4356 |
This theorem is referenced by: (None) |
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