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Theorem fidomdm 8800
Description: Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
fidomdm (𝐹 ∈ Fin → dom 𝐹𝐹)

Proof of Theorem fidomdm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dmresv 6056 . 2 dom (𝐹 ↾ V) = dom 𝐹
2 finresfin 8743 . . . 4 (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin)
3 fvex 6682 . . . . . . 7 (1st𝑥) ∈ V
4 eqid 2821 . . . . . . 7 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))
53, 4fnmpti 6490 . . . . . 6 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) Fn (𝐹 ↾ V)
6 dffn4 6595 . . . . . 6 ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) Fn (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
75, 6mpbi 232 . . . . 5 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))
8 relres 5881 . . . . . 6 Rel (𝐹 ↾ V)
9 reldm 7742 . . . . . 6 (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
10 foeq3 6587 . . . . . 6 (dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) → ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))))
118, 9, 10mp2b 10 . . . . 5 ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
127, 11mpbir 233 . . . 4 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)
13 fodomfi 8796 . . . 4 (((𝐹 ↾ V) ∈ Fin ∧ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)) → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
142, 12, 13sylancl 588 . . 3 (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
15 resss 5877 . . . 4 (𝐹 ↾ V) ⊆ 𝐹
16 ssdomg 8554 . . . 4 (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹))
1715, 16mpi 20 . . 3 (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹)
18 domtr 8561 . . 3 ((dom (𝐹 ↾ V) ≼ (𝐹 ↾ V) ∧ (𝐹 ↾ V) ≼ 𝐹) → dom (𝐹 ↾ V) ≼ 𝐹)
1914, 17, 18syl2anc 586 . 2 (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ 𝐹)
201, 19eqbrtrrid 5101 1 (𝐹 ∈ Fin → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1533  wcel 2110  Vcvv 3494  wss 3935   class class class wbr 5065  cmpt 5145  dom cdm 5554  ran crn 5555  cres 5556  Rel wrel 5559   Fn wfn 6349  ontowfo 6352  cfv 6354  1st c1st 7686  cdom 8506  Fincfn 8508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7580  df-1st 7688  df-2nd 7689  df-1o 8101  df-er 8288  df-en 8509  df-dom 8510  df-fin 8512
This theorem is referenced by:  dmfi  8801  hashfun  13797
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