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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 5260 | . 2 ⊢ dom ∅ = ∅ | |
2 | dm0rn0 5263 | . 2 ⊢ (dom ∅ = ∅ ↔ ran ∅ = ∅) | |
3 | 1, 2 | mpbi 219 | 1 ⊢ ran ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∅c0 3874 dom cdm 5038 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: ima0 5400 0ima 5401 rnxpid 5486 xpima 5495 f0 5999 2ndval 7062 frxp 7174 oarec 7529 fodomr 7996 dfac5lem3 8831 itunitc 9126 0rest 15913 arwval 16516 pmtrfrn 17701 psgnsn 17763 oppglsm 17880 mpfrcl 19339 ply1frcl 19504 nbgra0edg 25961 uvtx01vtx 26020 rusgra0edg 26482 0ngrp 26749 bafval 26843 locfinref 29236 esumrnmpt2 29457 sibf0 29723 mvtval 30651 mrsubrn 30664 mrsub0 30667 mrsubf 30668 mrsubccat 30669 mrsubcn 30670 mrsubco 30672 mrsubvrs 30673 elmsubrn 30679 msubrn 30680 msubf 30683 mstaval 30695 mzpmfp 36328 dmnonrel 36915 imanonrel 36918 conrel1d 36974 clsneibex 37420 neicvgbex 37430 sge00 39269 0grsubgr 40502 0uhgrsubgr 40503 |
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