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Theorem fidomdm 9372
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 6222 . 2 dom (𝐹 ↾ V) = dom 𝐹
2 finresfin 9302 . . . 4 (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin)
3 fvex 6920 . . . . . . 7 (1st𝑥) ∈ V
4 eqid 2735 . . . . . . 7 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))
53, 4fnmpti 6712 . . . . . 6 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) Fn (𝐹 ↾ V)
6 dffn4 6827 . . . . . 6 ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) Fn (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
75, 6mpbi 230 . . . . 5 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))
8 relres 6026 . . . . . 6 Rel (𝐹 ↾ V)
9 reldm 8068 . . . . . 6 (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
10 foeq3 6819 . . . . . 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 231 . . . 4 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)
13 fodomfi 9348 . . . 4 (((𝐹 ↾ V) ∈ Fin ∧ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)) → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
142, 12, 13sylancl 586 . . 3 (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
15 resss 6022 . . . 4 (𝐹 ↾ V) ⊆ 𝐹
16 ssdomg 9039 . . . 4 (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹))
1715, 16mpi 20 . . 3 (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹)
18 domtr 9046 . . 3 ((dom (𝐹 ↾ V) ≼ (𝐹 ↾ V) ∧ (𝐹 ↾ V) ≼ 𝐹) → dom (𝐹 ↾ V) ≼ 𝐹)
1914, 17, 18syl2anc 584 . 2 (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ 𝐹)
201, 19eqbrtrrid 5184 1 (𝐹 ∈ Fin → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  cres 5691  Rel wrel 5694   Fn wfn 6558  ontowfo 6561  cfv 6563  1st c1st 8011  cdom 8982  Fincfn 8984
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-en 8985  df-dom 8986  df-fin 8988
This theorem is referenced by:  dmfi  9373  hashfun  14473
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