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Mirrors > Home > MPE Home > Th. List > ima0 | Structured version Visualization version GIF version |
Description: Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.) |
Ref | Expression |
---|---|
ima0 | ⊢ (𝐴 “ ∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5051 | . 2 ⊢ (𝐴 “ ∅) = ran (𝐴 ↾ ∅) | |
2 | res0 5321 | . . 3 ⊢ (𝐴 ↾ ∅) = ∅ | |
3 | 2 | rneqi 5273 | . 2 ⊢ ran (𝐴 ↾ ∅) = ran ∅ |
4 | rn0 5298 | . 2 ⊢ ran ∅ = ∅ | |
5 | 1, 3, 4 | 3eqtri 2636 | 1 ⊢ (𝐴 “ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∅c0 3874 ran crn 5039 ↾ cres 5040 “ cima 5041 |
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-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: csbima12 5402 relimasn 5407 elimasni 5411 inisegn0 5416 dffv3 6099 supp0cosupp0 7221 imacosupp 7222 ecexr 7634 domunfican 8118 fodomfi 8124 efgrelexlema 17985 dprdsn 18258 cnindis 20906 cnhaus 20968 cmpfi 21021 xkouni 21212 xkoccn 21232 mbfima 23205 ismbf2d 23214 limcnlp 23448 mdeg0 23634 pserulm 23980 0pth 26100 spthispth 26103 1pthonlem2 26120 eupath2 26507 disjpreima 28779 imadifxp 28796 dstrvprob 29860 opelco3 30923 funpartlem 31219 poimirlem1 32580 poimirlem2 32581 poimirlem3 32582 poimirlem4 32583 poimirlem5 32584 poimirlem6 32585 poimirlem7 32586 poimirlem10 32589 poimirlem11 32590 poimirlem12 32591 poimirlem13 32592 poimirlem16 32595 poimirlem17 32596 poimirlem19 32598 poimirlem20 32599 poimirlem22 32601 poimirlem23 32602 poimirlem24 32603 poimirlem25 32604 poimirlem28 32607 poimirlem29 32608 poimirlem31 32610 he0 37098 smfresal 39673 sPthisPth 40932 pthdlem2 40974 0pth-av 41293 1pthdlem2 41303 eupth2lemb 41405 |
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