Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > brrelex12 | Structured version Visualization version GIF version |
Description: A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
brrelex12 | ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5045 | . . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 205 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 2 | ssbrd 4626 | . . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴(V × V)𝐵)) |
4 | 3 | imp 444 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴(V × V)𝐵) |
5 | brxp 5071 | . 2 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
6 | 4, 5 | sylib 207 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 × cxp 5036 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: brrelex 5080 brrelex2 5081 relbrcnvg 5423 ovprc 6581 oprabv 6601 brovex 7235 ersym 7641 relelec 7674 encv 7849 fsuppunbi 8179 fpwwe2lem2 9333 fpwwelem 9346 brfi1uzind 13135 brfi1uzindOLD 13141 isstruct2 15704 brssc 16297 cofuval2 16370 isfull 16393 isfth 16397 isnat 16430 pslem 17029 frgpuplem 18008 dvdsr 18469 ulmval 23938 perpln1 25405 perpln2 25406 iseupa 26492 opelco3 30923 rngoablo2 32878 aovprc 39917 aovrcl 39918 |
Copyright terms: Public domain | W3C validator |