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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 5059 | . 2 ⊢ (𝐴𝐶𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶) | |
2 | brelrn.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | brelrn.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | brelrn 5806 | . 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) |
5 | 1, 4 | sylbir 237 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3494 〈cop 4566 class class class wbr 5058 ran crn 5550 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-cnv 5557 df-dm 5559 df-rn 5560 |
This theorem is referenced by: dfres3 5852 zfrep6 7650 2ndrn 7734 disjen 8668 r0weon 9432 gsum2dlem1 19084 gsum2dlem2 19085 iss2 35595 rfovcnvf1od 40343 |
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