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Theorem fidomdm 8483
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 5807 . 2 dom (𝐹 ↾ V) = dom 𝐹
2 finresfin 8426 . . . 4 (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin)
3 fvex 6422 . . . . . . 7 (1st𝑥) ∈ V
4 eqid 2797 . . . . . . 7 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))
53, 4fnmpti 6231 . . . . . 6 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) Fn (𝐹 ↾ V)
6 dffn4 6335 . . . . . 6 ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) Fn (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
75, 6mpbi 222 . . . . 5 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))
8 relres 5634 . . . . . 6 Rel (𝐹 ↾ V)
9 reldm 7452 . . . . . 6 (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
10 foeq3 6327 . . . . . 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 223 . . . 4 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)
13 fodomfi 8479 . . . 4 (((𝐹 ↾ V) ∈ Fin ∧ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)) → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
142, 12, 13sylancl 581 . . 3 (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
15 resss 5630 . . . 4 (𝐹 ↾ V) ⊆ 𝐹
16 ssdomg 8239 . . . 4 (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹))
1715, 16mpi 20 . . 3 (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹)
18 domtr 8246 . . 3 ((dom (𝐹 ↾ V) ≼ (𝐹 ↾ V) ∧ (𝐹 ↾ V) ≼ 𝐹) → dom (𝐹 ↾ V) ≼ 𝐹)
1914, 17, 18syl2anc 580 . 2 (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ 𝐹)
201, 19syl5eqbrr 4877 1 (𝐹 ∈ Fin → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  Vcvv 3383  wss 3767   class class class wbr 4841  cmpt 4920  dom cdm 5310  ran crn 5311  cres 5312  Rel wrel 5315   Fn wfn 6094  ontowfo 6097  cfv 6099  1st c1st 7397  cdom 8191  Fincfn 8193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-om 7298  df-1st 7399  df-2nd 7400  df-1o 7797  df-er 7980  df-en 8194  df-dom 8195  df-fin 8197
This theorem is referenced by:  dmfi  8484  hashfun  13469
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