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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 5785 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
3 | df-br 5031 | . . 3 ⊢ (𝑥𝐵𝐴 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵) | |
4 | 3 | exbii 1849 | . 2 ⊢ (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
5 | 2, 4 | bitr4i 281 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 〈cop 4531 class class class wbr 5030 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: dmcosseq 5809 inisegn0 5928 rnco 6072 dffo4 6846 fvclss 6979 rntpos 7888 fpwwe2lem11 10051 fpwwe2lem12 10052 fclim 14902 perfdvf 24506 dftr6 33099 dffr5 33102 brsset 33463 dfon3 33466 brtxpsd 33468 dffix2 33479 elsingles 33492 dfrdg4 33525 undmrnresiss 40304 |
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