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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 2610 | . . 3 ⊢ ∅ = ∅ | |
2 | fn0 5924 | . . 3 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
3 | 1, 2 | mpbir 220 | . 2 ⊢ ∅ Fn ∅ |
4 | rn0 5298 | . . 3 ⊢ ran ∅ = ∅ | |
5 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
6 | 4, 5 | eqsstri 3598 | . 2 ⊢ ran ∅ ⊆ 𝐴 |
7 | df-f 5808 | . 2 ⊢ (∅:∅⟶𝐴 ↔ (∅ Fn ∅ ∧ ran ∅ ⊆ 𝐴)) | |
8 | 3, 6, 7 | mpbir2an 957 | 1 ⊢ ∅:∅⟶𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊆ wss 3540 ∅c0 3874 ran crn 5039 Fn wfn 5799 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: f00 6000 f0bi 6001 f10 6081 map0g 7783 ac6sfi 8089 oif 8318 wrd0 13185 0csh0 13390 ram0 15564 0ssc 16320 0subcat 16321 gsum0 17101 ga0 17554 0frgp 18015 ptcmpfi 21426 0met 21981 perfdvf 23473 uhgr0e 25737 uhgr0 25739 uhgra0 25838 umgra0 25854 vdgr0 26427 locfinref 29236 matunitlindf 32577 poimirlem28 32607 mapdm0 38378 0cnf 38762 dvnprodlem3 38838 mbf0 38849 sge00 39269 hoidmvlelem3 39487 griedg0prc 40488 |
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