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| Description: Membership in a range. (Contributed by NM, 2-Apr-2004.) | 
| Ref | Expression | 
|---|---|
| elrn.1 | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| elrn | ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elrn.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | elrng 5902 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∃wex 1779 ∈ 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: dmcosseq 5987 dmcosseqOLD 5988 inisegn0 6116 rnco 6272 dffo4 7123 fvclss 7261 rntpos 8264 fpwwe2lem10 10680 fpwwe2lem11 10681 fclim 15589 perfdvf 25938 dftr6 35751 dffr5 35754 brsset 35890 dfon3 35893 brtxpsd 35895 dffix2 35906 elsingles 35919 dfrdg4 35952 undmrnresiss 43617 | 
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