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Mirrors > Home > MPE Home > Th. List > opelrn | Structured version Visualization version 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 5149 | . 2 ⊢ (𝐴𝐶𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶) | |
2 | brelrn.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | brelrn.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | brelrn 5956 | . 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) |
5 | 1, 4 | sylbir 235 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 〈cop 4637 class class class wbr 5148 ran crn 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: dfres3 6005 relssdmrn 6290 zfrep6 7978 2ndrn 8065 disjen 9173 r0weon 10050 gsum2dlem1 20003 gsum2dlem2 20004 iss2 38326 rfovcnvf1od 43994 |
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