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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 5548 | . 2 ⊢ (𝐴 “ ∅) = ran (𝐴 ↾ ∅) | |
2 | res0 5839 | . . 3 ⊢ (𝐴 ↾ ∅) = ∅ | |
3 | 2 | rneqi 5790 | . 2 ⊢ ran (𝐴 ↾ ∅) = ran ∅ |
4 | rn0 5779 | . 2 ⊢ ran ∅ = ∅ | |
5 | 1, 3, 4 | 3eqtri 2766 | 1 ⊢ (𝐴 “ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∅c0 4221 ran crn 5536 ↾ cres 5537 “ cima 5538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-xp 5541 df-cnv 5543 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 |
This theorem is referenced by: csbima12 5931 relimasn 5936 elimasni 5940 inisegn0 5945 dffv3 6682 suppco 7913 supp0cosupp0 7915 ecexr 8337 imafi 8785 domunfican 8877 fodomfi 8882 efgrelexlema 19005 dprdsn 19289 cnindis 22055 cnhaus 22117 cmpfi 22171 xkouni 22362 xkoccn 22382 mbfima 24394 ismbf2d 24404 limcnlp 24642 mdeg0 24835 pserulm 25181 spthispth 27679 pthdlem2 27721 0pth 28074 1pthdlem2 28085 eupth2lemb 28186 disjpreima 30509 imadifxp 30526 2ndimaxp 30570 swrdrndisj 30816 gsumpart 30904 zarclsint 31406 dstrvprob 32020 opelco3 33335 old0 33698 made0 33712 negs0s 33779 funpartlem 33899 poimirlem1 35433 poimirlem2 35434 poimirlem3 35435 poimirlem4 35436 poimirlem5 35437 poimirlem6 35438 poimirlem7 35439 poimirlem10 35442 poimirlem11 35443 poimirlem12 35444 poimirlem13 35445 poimirlem16 35448 poimirlem17 35449 poimirlem19 35451 poimirlem20 35452 poimirlem22 35454 poimirlem23 35455 poimirlem24 35456 poimirlem25 35457 poimirlem28 35460 poimirlem29 35461 poimirlem31 35463 he0 40978 smfresal 43901 predisj 45735 |
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