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Theorem fidomdm 9374
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 6220 . 2 dom (𝐹 ↾ V) = dom 𝐹
2 finresfin 9304 . . . 4 (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin)
3 fvex 6919 . . . . . . 7 (1st𝑥) ∈ V
4 eqid 2737 . . . . . . 7 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))
53, 4fnmpti 6711 . . . . . 6 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) Fn (𝐹 ↾ V)
6 dffn4 6826 . . . . . 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 6023 . . . . . 6 Rel (𝐹 ↾ V)
9 reldm 8069 . . . . . 6 (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
10 foeq3 6818 . . . . . 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 9350 . . . 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 6019 . . . 4 (𝐹 ↾ V) ⊆ 𝐹
16 ssdomg 9040 . . . 4 (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹))
1715, 16mpi 20 . . 3 (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹)
18 domtr 9047 . . 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 5179 1 (𝐹 ∈ Fin → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686  cres 5687  Rel wrel 5690   Fn wfn 6556  ontowfo 6559  cfv 6561  1st c1st 8012  cdom 8983  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989
This theorem is referenced by:  dmfi  9375  hashfun  14476
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