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Mirrors > Home > MPE Home > Th. List > rn0 | Structured version Visualization version GIF version |
Description: The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
rn0 | ⊢ ran ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dm0 5783 | . 2 ⊢ dom ∅ = ∅ | |
2 | dm0rn0 5788 | . 2 ⊢ (dom ∅ = ∅ ↔ ran ∅ = ∅) | |
3 | 1, 2 | mpbi 231 | 1 ⊢ ran ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∅c0 4288 dom cdm 5548 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: ima0 5938 0ima 5939 rnxpid 6023 xpima 6032 f0 6553 rnfvprc 6657 2ndval 7681 frxp 7809 oarec 8177 fodomr 8656 dfac5lem3 9539 itunitc 9831 0rest 16691 arwval 17291 psgnsn 18577 oppglsm 18696 mpfrcl 20226 ply1frcl 20409 edgval 26761 0grsubgr 26987 0uhgrsubgr 26988 0ngrp 28215 bafval 28308 tocycf 30686 tocyc01 30687 locfinref 31004 esumrnmpt2 31226 sibf0 31491 mvtval 32644 mrsubvrs 32666 mstaval 32688 mzpmfp 39222 dmnonrel 39828 imanonrel 39831 conrel1d 39886 clsneibex 40330 neicvgbex 40340 sge00 42535 |
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