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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 2737 | . . 3 ⊢ ∅ = ∅ | |
| 2 | fn0 6699 | . . 3 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
| 3 | 1, 2 | mpbir 231 | . 2 ⊢ ∅ Fn ∅ |
| 4 | rn0 5936 | . . 3 ⊢ ran ∅ = ∅ | |
| 5 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
| 6 | 4, 5 | eqsstri 4030 | . 2 ⊢ ran ∅ ⊆ 𝐴 |
| 7 | df-f 6565 | . 2 ⊢ (∅:∅⟶𝐴 ↔ (∅ Fn ∅ ∧ ran ∅ ⊆ 𝐴)) | |
| 8 | 3, 6, 7 | mpbir2an 711 | 1 ⊢ ∅:∅⟶𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ⊆ wss 3951 ∅c0 4333 ran crn 5686 Fn wfn 6556 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: f00 6790 f0bi 6791 f10 6881 map0g 8924 ac6sfi 9320 oif 9570 wrd0 14577 0csh0 14831 ram0 17060 0ssc 17882 0subcat 17883 setc2ohom 18140 cat1lem 18141 gsum0 18697 ga0 19316 0frgp 19797 ptcmpfi 23821 0met 24376 perfdvf 25938 uhgr0e 29088 uhgr0 29090 griedg0prc 29281 locfinref 33840 matunitlindf 37625 poimirlem28 37655 sticksstones11 42157 climlimsupcex 45784 0cnf 45892 dvnprodlem3 45963 sge00 46391 hoidmvlelem3 46612 |
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