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Mirrors > Home > MPE Home > Th. List > elrn2 | Structured version Visualization version GIF version |
Description: Membership in a range. (Contributed by NM, 10-Jul-1994.) |
Ref | Expression |
---|---|
elrn.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elrn2 | ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elrn2g 5890 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1780 ∈ wcel 2105 Vcvv 3473 〈cop 4634 ran crn 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 |
This theorem is referenced by: dmrnssfld 5969 rniun 6147 ssrnres 6177 relssdmrnOLD 6268 fvelrn 7078 tz7.48-1 8447 prsrn 33194 dfrn5 35050 funressndmafv2rn 46230 |
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