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| 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 5187 | . 2 ⊢ (𝐴𝑅𝐵 → 𝐴(𝐶 × 𝐷)𝐵) | 
| 3 | brxp 5733 | . 2 ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
| 4 | 2, 3 | sylib 218 | 1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ⊆ wss 3950 class class class wbr 5142 × cxp 5682 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 | 
| This theorem is referenced by: brab2a 5778 soirri 6145 sotri 6146 sotri2 6148 sotri3 6149 ndmovord 7624 ndmovordi 7625 swoer 8777 brecop2 8852 ecopovsym 8860 ecopovtrn 8861 hartogslem1 9583 nlt1pi 10947 indpi 10948 nqerf 10971 ordpipq 10983 lterpq 11011 ltexnq 11016 ltbtwnnq 11019 ltrnq 11020 prnmadd 11038 genpcd 11047 nqpr 11055 1idpr 11070 ltexprlem4 11080 ltexpri 11084 ltaprlem 11085 prlem936 11088 reclem2pr 11089 reclem3pr 11090 reclem4pr 11091 suplem1pr 11093 suplem2pr 11094 supexpr 11095 recexsrlem 11144 addgt0sr 11145 mulgt0sr 11146 mappsrpr 11149 map2psrpr 11151 supsrlem 11152 supsr 11153 ltresr 11181 dfle2 13190 dflt2 13191 dvdszrcl 16296 letsr 18639 hmphtop 23787 brtxp2 35883 brpprod3a 35888 brxrn2 38377 aks6d1c1p1rcl 42110 iccdisj2 48801 i0oii 48824 io1ii 48825 | 
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