Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5077 | . 2 ⊢ (𝐴𝐶𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐶) | |
2 | brelrn.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | brelrn.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | brelrn 5853 | . 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) |
5 | 1, 4 | sylbir 234 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ran 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3431 〈cop 4569 class class class wbr 5076 ran crn 5592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pr 5354 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-br 5077 df-opab 5139 df-cnv 5599 df-dm 5601 df-rn 5602 |
This theorem is referenced by: dfres3 5898 zfrep6 7797 2ndrn 7882 disjen 8919 r0weon 9766 gsum2dlem1 19569 gsum2dlem2 19570 iss2 36476 rfovcnvf1od 41582 |
Copyright terms: Public domain | W3C validator |