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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 6060 | . . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅ | |
2 | rn0 5917 | . . 3 ⊢ ran ∅ = ∅ | |
3 | 1, 2 | sseqtri 4014 | . 2 ⊢ (∅ “ 𝐴) ⊆ ∅ |
4 | 0ss 4392 | . 2 ⊢ ∅ ⊆ (∅ “ 𝐴) | |
5 | 3, 4 | eqssi 3994 | 1 ⊢ (∅ “ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∅c0 4318 ran crn 5670 “ cima 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 |
This theorem is referenced by: csbrn 6191 nghmfval 24168 isnghm 24169 mptiffisupp 31786 mthmval 34397 ec0 37043 0he 42304 limsup0 44183 0cnf 44366 |
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