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| Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.) | 
| Ref | Expression | 
|---|---|
| brelrn.1 | ⊢ 𝐴 ∈ V | 
| brelrn.2 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| brelrn | ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | brelrn.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | brelrn.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | brelrng 5952 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶) | |
| 4 | 1, 2, 3 | mp3an12 1453 | 1 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ran crn 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 | 
| This theorem is referenced by: opelrn 5954 dfco2a 6266 cores 6269 dffun9 6595 funcnv 6635 rntpos 8264 rnttrcl 9762 aceq3lem 10160 axdclem 10559 axdclem2 10560 cotr2g 15015 shftfval 15109 psdmrn 18618 metustexhalf 24569 itg1addlem4 25734 | 
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