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Mirrors > Home > ILE Home > Th. List > elrn | GIF version |
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 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | elrn2 4781 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
3 | df-br 3930 | . . 3 ⊢ (𝑥𝐵𝐴 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵) | |
4 | 3 | exbii 1584 | . 2 ⊢ (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
5 | 2, 4 | bitr4i 186 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 〈cop 3530 class class class wbr 3929 ran crn 4540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: dmcosseq 4810 rnco 5045 dffo4 5568 rntpos 6154 fclim 11063 dvfgg 12826 |
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