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| 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 5930 | . 2 ⊢ dom ∅ = ∅ | |
| 2 | dm0rn0 5934 | . 2 ⊢ (dom ∅ = ∅ ↔ ran ∅ = ∅) | |
| 3 | 1, 2 | mpbi 230 | 1 ⊢ ran ∅ = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∅c0 4332 dom cdm 5684 ran crn 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 | 
| This theorem is referenced by: ima0 6094 0ima 6095 rnxpid 6192 xpima 6201 f0 6788 rnfvprc 6899 2ndval 8018 frxp 8152 oarec 8601 fodomr 9169 fodomfir 9369 dfac5lem3 10166 itunitc 10462 relexprnd 15088 0rest 17475 arwval 18089 psgnsn 19539 oppglsm 19661 mpfrcl 22110 ply1frcl 22323 edgval 29067 0grsubgr 29296 0uhgrsubgr 29297 0ngrp 30531 bafval 30624 tocycf 33138 tocyc01 33139 unitprodclb 33418 locfinref 33841 esumrnmpt2 34070 sibf0 34337 mvtval 35506 mrsubvrs 35528 mstaval 35550 mzpmfp 42763 dmnonrel 43608 imanonrel 43611 conrel1d 43681 clsneibex 44120 neicvgbex 44130 sge00 46396 | 
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