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Mirrors > Home > MPE Home > Th. List > fn0 | Structured version Visualization version GIF version |
Description: A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fn0 | ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 6670 | . . 3 ⊢ (𝐹 Fn ∅ → Rel 𝐹) | |
2 | fndm 6671 | . . 3 ⊢ (𝐹 Fn ∅ → dom 𝐹 = ∅) | |
3 | reldm0 5940 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅)) | |
4 | 3 | biimpar 477 | . . 3 ⊢ ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅) |
5 | 1, 2, 4 | syl2anc 584 | . 2 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅) |
6 | fun0 6632 | . . . 4 ⊢ Fun ∅ | |
7 | dm0 5933 | . . . 4 ⊢ dom ∅ = ∅ | |
8 | df-fn 6565 | . . . 4 ⊢ (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅)) | |
9 | 6, 7, 8 | mpbir2an 711 | . . 3 ⊢ ∅ Fn ∅ |
10 | fneq1 6659 | . . 3 ⊢ (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅)) | |
11 | 9, 10 | mpbiri 258 | . 2 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅) |
12 | 5, 11 | impbii 209 | 1 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∅c0 4338 dom cdm 5688 Rel wrel 5693 Fun wfun 6556 Fn wfn 6557 |
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-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-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-fun 6564 df-fn 6565 |
This theorem is referenced by: mpt0 6710 f0 6789 f00 6790 f0bi 6791 f1o00 6883 fo00 6884 tpos0 8279 ixp0x 8964 0fz1 13580 hashf1 14492 fuchom 18016 fuchomOLD 18017 grpinvfvi 19012 mulgfval 19099 mulgfvalALT 19100 mulgfvi 19103 0frgp 19811 invrfval 20405 psrvscafval 21985 tmdgsum 24118 deg1fvi 26138 hon0 31821 fnchoice 44966 dvnprodlem3 45903 |
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