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Theorem fidomdm 8187
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 5552 . 2 dom (𝐹 ↾ V) = dom 𝐹
2 finresfin 8130 . . . 4 (𝐹 ∈ Fin → (𝐹 ↾ V) ∈ Fin)
3 fvex 6158 . . . . . . 7 (1st𝑥) ∈ V
4 eqid 2621 . . . . . . 7 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) = (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))
53, 4fnmpti 5979 . . . . . 6 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) Fn (𝐹 ↾ V)
6 dffn4 6078 . . . . . 6 ((𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)) Fn (𝐹 ↾ V) ↔ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
75, 6mpbi 220 . . . . 5 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥))
8 relres 5385 . . . . . 6 Rel (𝐹 ↾ V)
9 reldm 7164 . . . . . 6 (Rel (𝐹 ↾ V) → dom (𝐹 ↾ V) = ran (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)))
10 foeq3 6070 . . . . . 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 221 . . . 4 (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)
13 fodomfi 8183 . . . 4 (((𝐹 ↾ V) ∈ Fin ∧ (𝑥 ∈ (𝐹 ↾ V) ↦ (1st𝑥)):(𝐹 ↾ V)–onto→dom (𝐹 ↾ V)) → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
142, 12, 13sylancl 693 . . 3 (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ (𝐹 ↾ V))
15 resss 5381 . . . 4 (𝐹 ↾ V) ⊆ 𝐹
16 ssdomg 7945 . . . 4 (𝐹 ∈ Fin → ((𝐹 ↾ V) ⊆ 𝐹 → (𝐹 ↾ V) ≼ 𝐹))
1715, 16mpi 20 . . 3 (𝐹 ∈ Fin → (𝐹 ↾ V) ≼ 𝐹)
18 domtr 7953 . . 3 ((dom (𝐹 ↾ V) ≼ (𝐹 ↾ V) ∧ (𝐹 ↾ V) ≼ 𝐹) → dom (𝐹 ↾ V) ≼ 𝐹)
1914, 17, 18syl2anc 692 . 2 (𝐹 ∈ Fin → dom (𝐹 ↾ V) ≼ 𝐹)
201, 19syl5eqbrr 4649 1 (𝐹 ∈ Fin → dom 𝐹𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555   class class class wbr 4613  cmpt 4673  dom cdm 5074  ran crn 5075  cres 5076  Rel wrel 5079   Fn wfn 5842  ontowfo 5845  cfv 5847  1st c1st 7111  cdom 7897  Fincfn 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1st 7113  df-2nd 7114  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903
This theorem is referenced by:  dmfi  8188  hashfun  13164
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