Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5365 | . 2 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
3 | df-br 4654 | . . 3 ⊢ (𝑥𝐵𝐴 ↔ 〈𝑥, 𝐴〉 ∈ 𝐵) | |
4 | 3 | exbii 1774 | . 2 ⊢ (∃𝑥 𝑥𝐵𝐴 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) |
5 | 2, 4 | bitr4i 267 | 1 ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 〈cop 4183 class class class wbr 4653 ran crn 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: dmcosseq 5387 inisegn0 5497 rnco 5641 dffo4 6375 fvclss 6500 rntpos 7365 fpwwe2lem11 9462 fpwwe2lem12 9463 fclim 14284 perfdvf 23667 dftr6 31640 dffr5 31643 brsset 31996 dfon3 31999 brtxpsd 32001 dffix2 32012 elsingles 32025 dfrdg4 32058 undmrnresiss 37910 |
Copyright terms: Public domain | W3C validator |