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Mirrors > Home > ILE Home > Th. List > brrelex2i | GIF version |
Description: The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
brrelexi.1 | ⊢ Rel 𝑅 |
Ref | Expression |
---|---|
brrelex2i | ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brrelexi.1 | . 2 ⊢ Rel 𝑅 | |
2 | brrelex2 4663 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) | |
3 | 1, 2 | mpan 424 | 1 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 class class class wbr 4000 Rel wrel 4627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4628 df-rel 4629 |
This theorem is referenced by: vtoclr 4670 brdomi 6742 xpdom2 6824 xpdom1g 6826 mapdom1g 6840 djudom 7085 difinfsn 7092 enomnilem 7129 enmkvlem 7152 enwomnilem 7160 djuenun 7204 aprcl 8580 hashinfom 10729 clim 11260 ntrivcvgap0 11528 |
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