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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 5934 | . 2 ⊢ dom ∅ = ∅ | |
2 | dm0rn0 5938 | . 2 ⊢ (dom ∅ = ∅ ↔ ran ∅ = ∅) | |
3 | 1, 2 | mpbi 230 | 1 ⊢ ran ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∅c0 4339 dom cdm 5689 ran crn 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: ima0 6097 0ima 6098 rnxpid 6195 xpima 6204 f0 6790 rnfvprc 6901 2ndval 8016 frxp 8150 oarec 8599 fodomr 9167 fodomfir 9366 dfac5lem3 10163 itunitc 10459 relexprnd 15084 0rest 17476 arwval 18097 psgnsn 19553 oppglsm 19675 mpfrcl 22127 ply1frcl 22338 edgval 29081 0grsubgr 29310 0uhgrsubgr 29311 0ngrp 30540 bafval 30633 tocycf 33120 tocyc01 33121 unitprodclb 33397 locfinref 33802 esumrnmpt2 34049 sibf0 34316 mvtval 35485 mrsubvrs 35507 mstaval 35529 mzpmfp 42735 dmnonrel 43580 imanonrel 43583 conrel1d 43653 clsneibex 44092 neicvgbex 44102 sge00 46332 |
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