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Mirrors > Home > MPE Home > Th. List > elrn | Structured version Visualization version 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 5286 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
3 | df-br 4584 | . . 3 ⊢ (𝑥𝐵𝐴 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵) | |
4 | 3 | exbii 1764 | . 2 ⊢ (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
5 | 2, 4 | bitr4i 266 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: dmcosseq 5308 inisegn0 5416 rnco 5558 dffo4 6283 fvclss 6404 rntpos 7252 fpwwe2lem11 9341 fpwwe2lem12 9342 fclim 14132 perfdvf 23473 dftr6 30893 dffr5 30896 brsset 31166 dfon3 31169 brtxpsd 31171 dffix2 31182 elsingles 31195 dfrdg4 31228 undmrnresiss 36929 |
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