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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 5355 | . 2 ⊢ (𝐴 “ ∅) = ran (𝐴 ↾ ∅) | |
2 | res0 5633 | . . 3 ⊢ (𝐴 ↾ ∅) = ∅ | |
3 | 2 | rneqi 5584 | . 2 ⊢ ran (𝐴 ↾ ∅) = ran ∅ |
4 | rn0 5610 | . 2 ⊢ ran ∅ = ∅ | |
5 | 1, 3, 4 | 3eqtri 2853 | 1 ⊢ (𝐴 “ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∅c0 4144 ran crn 5343 ↾ cres 5344 “ cima 5345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-xp 5348 df-cnv 5350 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 |
This theorem is referenced by: csbima12 5724 relimasn 5729 elimasni 5733 inisegn0 5738 dffv3 6429 supp0cosupp0 7599 imacosupp 7600 ecexr 8014 domunfican 8502 fodomfi 8508 efgrelexlema 18515 dprdsn 18789 cnindis 21467 cnhaus 21529 cmpfi 21582 xkouni 21773 xkoccn 21793 mbfima 23796 ismbf2d 23806 limcnlp 24041 mdeg0 24229 pserulm 24575 spthispth 27028 pthdlem2 27070 0pth 27501 1pthdlem2 27512 eupth2lemb 27614 disjpreima 29944 imadifxp 29961 dstrvprob 31079 opelco3 32216 funpartlem 32588 poimirlem1 33954 poimirlem2 33955 poimirlem3 33956 poimirlem4 33957 poimirlem5 33958 poimirlem6 33959 poimirlem7 33960 poimirlem10 33963 poimirlem11 33964 poimirlem12 33965 poimirlem13 33966 poimirlem16 33969 poimirlem17 33970 poimirlem19 33972 poimirlem20 33973 poimirlem22 33975 poimirlem23 33976 poimirlem24 33977 poimirlem25 33978 poimirlem28 33981 poimirlem29 33982 poimirlem31 33984 he0 38918 smfresal 41789 |
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