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Mirrors > Home > MPE Home > Th. List > 0fv | Structured version Visualization version GIF version |
Description: Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.) |
Ref | Expression |
---|---|
0fv | ⊢ (∅‘𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4119 | . . 3 ⊢ ¬ 𝐴 ∈ ∅ | |
2 | dm0 5542 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | 2 | eleq2i 2870 | . . 3 ⊢ (𝐴 ∈ dom ∅ ↔ 𝐴 ∈ ∅) |
4 | 1, 3 | mtbir 315 | . 2 ⊢ ¬ 𝐴 ∈ dom ∅ |
5 | ndmfv 6441 | . 2 ⊢ (¬ 𝐴 ∈ dom ∅ → (∅‘𝐴) = ∅) | |
6 | 4, 5 | ax-mp 5 | 1 ⊢ (∅‘𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1653 ∈ wcel 2157 ∅c0 4115 dom cdm 5312 ‘cfv 6101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-nul 4983 ax-pow 5035 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-dm 5322 df-iota 6064 df-fv 6109 |
This theorem is referenced by: fv2prc 6452 csbfv12 6455 0ov 6914 csbov123 6919 csbov 6920 elovmpt3imp 7124 bropopvvv 7492 bropfvvvvlem 7493 itunisuc 9529 itunitc1 9530 ccat1st1st 13651 str0 16236 ressbas 16255 cntrval 18064 cntzval 18066 cntzrcl 18072 sralem 19500 srasca 19504 sravsca 19505 sraip 19506 rlmval 19514 opsrle 19798 opsrbaslem 19800 mpfrcl 19840 evlval 19846 psr1val 19878 vr1val 19884 chrval 20195 ocvval 20336 elocv 20337 iscnp2 21372 resvsca 30346 mrsubfval 31922 msubfval 31938 poimirlem28 33926 0cnv 40718 |
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