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Mirrors > Home > MPE Home > Th. List > brel | Structured version Visualization version GIF version |
Description: Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
brel.1 | ⊢ 𝑅 ⊆ (𝐶 × 𝐷) |
Ref | Expression |
---|---|
brel | ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brel.1 | . . 3 ⊢ 𝑅 ⊆ (𝐶 × 𝐷) | |
2 | 1 | ssbri 5151 | . 2 ⊢ (𝐴𝑅𝐵 → 𝐴(𝐶 × 𝐷)𝐵) |
3 | brxp 5682 | . 2 ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
4 | 2, 3 | sylib 217 | 1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3911 class class class wbr 5106 × cxp 5632 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 |
This theorem is referenced by: brab2a 5726 soirri 6081 sotri 6082 sotri2 6084 sotri3 6085 ndmovord 7545 ndmovordi 7546 swoer 8681 brecop2 8753 ecopovsym 8761 ecopovtrn 8762 hartogslem1 9483 nlt1pi 10847 indpi 10848 nqerf 10871 ordpipq 10883 lterpq 10911 ltexnq 10916 ltbtwnnq 10919 ltrnq 10920 prnmadd 10938 genpcd 10947 nqpr 10955 1idpr 10970 ltexprlem4 10980 ltexpri 10984 ltaprlem 10985 prlem936 10988 reclem2pr 10989 reclem3pr 10990 reclem4pr 10991 suplem1pr 10993 suplem2pr 10994 supexpr 10995 recexsrlem 11044 addgt0sr 11045 mulgt0sr 11046 mappsrpr 11049 map2psrpr 11051 supsrlem 11052 supsr 11053 ltresr 11081 dfle2 13072 dflt2 13073 dvdszrcl 16146 letsr 18487 hmphtop 23145 brtxp2 34512 brpprod3a 34517 brxrn2 36883 iccdisj2 47016 i0oii 47038 io1ii 47039 |
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