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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 5058 | . 2 ⊢ (𝐴𝐶𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶) | |
2 | brelrn.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | brelrn.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | brelrn 5805 | . 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) |
5 | 1, 4 | sylbir 236 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 〈cop 4563 class class class wbr 5057 ran crn 5549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: dfres3 5851 zfrep6 7645 2ndrn 7729 disjen 8662 r0weon 9426 gsum2dlem1 19019 gsum2dlem2 19020 iss2 35482 rfovcnvf1od 40228 |
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