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Mirrors > Home > MPE Home > Th. List > f0 | Structured version Visualization version GIF version |
Description: The empty function. (Contributed by NM, 14-Aug-1999.) |
Ref | Expression |
---|---|
f0 | ⊢ ∅:∅⟶𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ ∅ = ∅ | |
2 | fn0 6699 | . . 3 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
3 | 1, 2 | mpbir 231 | . 2 ⊢ ∅ Fn ∅ |
4 | rn0 5938 | . . 3 ⊢ ran ∅ = ∅ | |
5 | 0ss 4405 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
6 | 4, 5 | eqsstri 4029 | . 2 ⊢ ran ∅ ⊆ 𝐴 |
7 | df-f 6566 | . 2 ⊢ (∅:∅⟶𝐴 ↔ (∅ Fn ∅ ∧ ran ∅ ⊆ 𝐴)) | |
8 | 3, 6, 7 | mpbir2an 711 | 1 ⊢ ∅:∅⟶𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⊆ wss 3962 ∅c0 4338 ran crn 5689 Fn wfn 6557 ⟶wf 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-fun 6564 df-fn 6565 df-f 6566 |
This theorem is referenced by: f00 6790 f0bi 6791 f10 6881 map0g 8922 ac6sfi 9317 oif 9567 wrd0 14573 0csh0 14827 ram0 17055 0ssc 17887 0subcat 17888 setc2ohom 18148 cat1lem 18149 gsum0 18709 ga0 19328 0frgp 19811 ptcmpfi 23836 0met 24391 perfdvf 25952 uhgr0e 29102 uhgr0 29104 griedg0prc 29295 locfinref 33801 matunitlindf 37604 poimirlem28 37634 sticksstones11 42137 climlimsupcex 45724 0cnf 45832 dvnprodlem3 45903 sge00 46331 hoidmvlelem3 46552 |
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