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Mirrors > Home > MPE Home > Th. List > 0ima | Structured version Visualization version GIF version |
Description: Image under the empty relation. (Contributed by FL, 11-Jan-2007.) |
Ref | Expression |
---|---|
0ima | ⊢ (∅ “ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 5783 | . . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅ | |
2 | rn0 5677 | . . 3 ⊢ ran ∅ = ∅ | |
3 | 1, 2 | sseqtri 3895 | . 2 ⊢ (∅ “ 𝐴) ⊆ ∅ |
4 | 0ss 4237 | . 2 ⊢ ∅ ⊆ (∅ “ 𝐴) | |
5 | 3, 4 | eqssi 3876 | 1 ⊢ (∅ “ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∅c0 4180 ran crn 5409 “ cima 5411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pr 5187 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-br 4931 df-opab 4993 df-xp 5414 df-cnv 5416 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 |
This theorem is referenced by: csbrn 5901 nghmfval 23037 isnghm 23038 mthmval 32342 ec0 35066 0he 39491 limsup0 41407 0cnf 41591 |
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