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Mirrors > Home > MPE Home > Th. List > fidomdm | Structured version Visualization version GIF version |
Description: Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
fidomdm | ⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmresv 5751 | . 2 ⊢ dom (𝐹 ↾ V) = dom 𝐹 | |
2 | finresfin 8351 | . . . 4 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin) | |
3 | fvex 6362 | . . . . . . 7 ⊢ (1st ‘𝑥) ∈ V | |
4 | eqid 2760 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) | |
5 | 3, 4 | fnmpti 6183 | . . . . . 6 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐹 ↾ V) |
6 | dffn4 6282 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) Fn (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))) | |
7 | 5, 6 | mpbi 220 | . . . . 5 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) |
8 | relres 5584 | . . . . . 6 ⊢ Rel (𝐹 ↾ V) | |
9 | reldm 7386 | . . . . . 6 ⊢ (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))) | |
10 | foeq3 6274 | . . . . . 6 ⊢ (dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)) → ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)))) | |
11 | 8, 9, 10 | mp2b 10 | . . . . 5 ⊢ ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥))) |
12 | 7, 11 | mpbir 221 | . . . 4 ⊢ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V) |
13 | fodomfi 8404 | . . . 4 ⊢ (((𝐹 ↾ V) ∈ Fin ∧ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st ‘𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)) → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V)) | |
14 | 2, 12, 13 | sylancl 697 | . . 3 ⊢ (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V)) |
15 | resss 5580 | . . . 4 ⊢ (𝐹 ↾ V) ⊆ 𝐹 | |
16 | ssdomg 8167 | . . . 4 ⊢ (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹)) | |
17 | 15, 16 | mpi 20 | . . 3 ⊢ (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹) |
18 | domtr 8174 | . . 3 ⊢ ((dom (𝐹 ↾ V) ≼ (𝐹 ↾ V) ∧ (𝐹 ↾ V) ≼ 𝐹) → dom (𝐹 ↾ V) ≼ 𝐹) | |
19 | 14, 17, 18 | syl2anc 696 | . 2 ⊢ (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ 𝐹) |
20 | 1, 19 | syl5eqbrr 4840 | 1 ⊢ (𝐹 ∈ Fin → dom 𝐹 ≼ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ⊆ wss 3715 class class class wbr 4804 ↦ cmpt 4881 dom cdm 5266 ran crn 5267 ↾ cres 5268 Rel wrel 5271 Fn wfn 6044 –onto→wfo 6047 ‘cfv 6049 1st c1st 7331 ≼ cdom 8119 Fincfn 8121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-om 7231 df-1st 7333 df-2nd 7334 df-1o 7729 df-er 7911 df-en 8122 df-dom 8123 df-fin 8125 |
This theorem is referenced by: dmfi 8409 hashfun 13416 |
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