|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 4337 | . . 3 ⊢ ¬ 𝐴 ∈ ∅ | |
| 2 | dm0 5930 | . . . 4 ⊢ dom ∅ = ∅ | |
| 3 | 2 | eleq2i 2832 | . . 3 ⊢ (𝐴 ∈ dom ∅ ↔ 𝐴 ∈ ∅) | 
| 4 | 1, 3 | mtbir 323 | . 2 ⊢ ¬ 𝐴 ∈ dom ∅ | 
| 5 | ndmfv 6940 | . 2 ⊢ (¬ 𝐴 ∈ dom ∅ → (∅‘𝐴) = ∅) | |
| 6 | 4, 5 | ax-mp 5 | 1 ⊢ (∅‘𝐴) = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2107 ∅c0 4332 dom cdm 5684 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-dm 5694 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: fv2prc 6950 csbfv12 6953 0ov 7469 elfvov1 7474 elfvov2 7475 csbov123 7476 csbov 7477 elovmpt3imp 7691 bropopvvv 8116 bropfvvvvlem 8117 itunisuc 10460 ccat1st1st 14667 str0 17227 ressbasOLD 17282 cntrval 19338 cntzval 19340 cntzrcl 19346 rlmval 21199 chrval 21539 ocvval 21686 elocv 21687 opsrle 22066 opsrbaslem 22068 opsrbaslemOLD 22069 mpfrcl 22110 evlval 22120 psr1val 22188 vr1val 22194 iscnp2 23248 resvsca 33357 mrsubfval 35514 msubfval 35530 poimirlem28 37656 0cnv 45762 elfvne0 48763 | 
| Copyright terms: Public domain | W3C validator |