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| Mirrors > Home > MPE Home > Th. List > brrelex2 | Structured version Visualization version GIF version | ||
| Description: If two classes are related by a binary relation, then the second class is a set. (Contributed by Mario Carneiro, 26-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| brrelex2 | ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | brrelex12 5737 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
| 2 | 1 | simprd 495 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 Rel wrel 5690 | 
| 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-ral 3062 df-rex 3071 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-xp 5691 df-rel 5692 | 
| This theorem is referenced by: brrelex2i 5742 releldm 5955 relelrn 5956 elrelimasn 6104 funbrfv 6957 relbrtpos 8262 ertr 8760 erth 8796 fsuppss 9423 pslem 18617 opeldifid 32612 eqvreltr 38608 eqvrelth 38612 frege124d 43774 frege133d 43778 climfv 45706 funbrafv2 47259 | 
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