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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 5814 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
3 | df-br 5058 | . . 3 ⊢ (𝑥𝐵𝐴 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵) | |
4 | 3 | exbii 1839 | . 2 ⊢ (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
5 | 2, 4 | bitr4i 279 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∃wex 1771 ∈ wcel 2105 Vcvv 3492 〈cop 4563 class class class wbr 5057 ran crn 5549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: dmcosseq 5837 inisegn0 5954 rnco 6098 dffo4 6861 fvclss 6992 rntpos 7894 fpwwe2lem11 10050 fpwwe2lem12 10051 fclim 14898 perfdvf 24428 dftr6 32883 dffr5 32886 brsset 33247 dfon3 33250 brtxpsd 33252 dffix2 33263 elsingles 33276 dfrdg4 33309 undmrnresiss 39842 |
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