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Theorem fidomdm 9328
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 β†’ dom 𝐹 β‰Ό 𝐹)
Proof of Theorem fidomdm
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 dmresv 6192 . 2 dom (𝐹 β†Ύ V) = dom 𝐹
2 finresfin 9269 . . . 4 (𝐹 ∈ Fin β†’ (𝐹 β†Ύ V) ∈ Fin)
3 fvex 6897 . . . . . . 7 (1st β€˜π‘₯) ∈ V
4 eqid 2726 . . . . . . 7 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) = (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯))
53, 4fnmpti 6686 . . . . . 6 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) Fn (𝐹 β†Ύ V)
6 dffn4 6804 . . . . . 6 ((π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) Fn (𝐹 β†Ύ V) ↔ (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)):(𝐹 β†Ύ V)–ontoβ†’ran (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)))
75, 6mpbi 229 . . . . 5 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)):(𝐹 β†Ύ V)–ontoβ†’ran (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯))
8 relres 6003 . . . . . 6 Rel (𝐹 β†Ύ V)
9 reldm 8026 . . . . . 6 (Rel (𝐹 β†Ύ V) β†’ dom (𝐹 β†Ύ V) = ran (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)))
10 foeq3 6796 . . . . . 6 (dom (𝐹 β†Ύ V) = ran (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) β†’ ((π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)):(𝐹 β†Ύ V)–ontoβ†’dom (𝐹 β†Ύ V) ↔ (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)):(𝐹 β†Ύ V)–ontoβ†’ran (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯))))
118, 9, 10mp2b 10 . . . . 5 ((π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)):(𝐹 β†Ύ V)–ontoβ†’dom (𝐹 β†Ύ V) ↔ (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)):(𝐹 β†Ύ V)–ontoβ†’ran (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)))
127, 11mpbir 230 . . . 4 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)):(𝐹 β†Ύ V)–ontoβ†’dom (𝐹 β†Ύ V)
13 fodomfi 9324 . . . 4 (((𝐹 β†Ύ V) ∈ Fin ∧ (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)):(𝐹 β†Ύ V)–ontoβ†’dom (𝐹 β†Ύ V)) β†’ dom (𝐹 β†Ύ V) β‰Ό (𝐹 β†Ύ V))
142, 12, 13sylancl 585 . . 3 (𝐹 ∈ Fin β†’ dom (𝐹 β†Ύ V) β‰Ό (𝐹 β†Ύ V))
15 resss 5999 . . . 4 (𝐹 β†Ύ V) βŠ† 𝐹
16 ssdomg 8995 . . . 4 (𝐹 ∈ Fin β†’ ((𝐹 β†Ύ V) βŠ† 𝐹 β†’ (𝐹 β†Ύ V) β‰Ό 𝐹))
1715, 16mpi 20 . . 3 (𝐹 ∈ Fin β†’ (𝐹 β†Ύ V) β‰Ό 𝐹)
18 domtr 9002 . . 3 ((dom (𝐹 β†Ύ V) β‰Ό (𝐹 β†Ύ V) ∧ (𝐹 β†Ύ V) β‰Ό 𝐹) β†’ dom (𝐹 β†Ύ V) β‰Ό 𝐹)
1914, 17, 18syl2anc 583 . 2 (𝐹 ∈ Fin β†’ dom (𝐹 β†Ύ V) β‰Ό 𝐹)
201, 19eqbrtrrid 5177 1 (𝐹 ∈ Fin β†’ dom 𝐹 β‰Ό 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3468   βŠ† wss 3943   class class class wbr 5141   ↦ cmpt 5224  dom cdm 5669  ran crn 5670   β†Ύ cres 5671  Rel wrel 5674   Fn wfn 6531  β€“ontoβ†’wfo 6534  β€˜cfv 6536  1st c1st 7969   β‰Ό cdom 8936  Fincfn 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8464  df-er 8702  df-en 8939  df-dom 8940  df-fin 8942
This theorem is referenced by:  dmfi  9329  hashfun  14400
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