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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 6199 . 2 dom (𝐹 β†Ύ V) = dom 𝐹
2 finresfin 9269 . . . 4 (𝐹 ∈ Fin β†’ (𝐹 β†Ύ V) ∈ Fin)
3 fvex 6904 . . . . . . 7 (1st β€˜π‘₯) ∈ V
4 eqid 2732 . . . . . . 7 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) = (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯))
53, 4fnmpti 6693 . . . . . 6 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) Fn (𝐹 β†Ύ V)
6 dffn4 6811 . . . . . 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 6010 . . . . . 6 Rel (𝐹 β†Ύ V)
9 reldm 8029 . . . . . 6 (Rel (𝐹 β†Ύ V) β†’ dom (𝐹 β†Ύ V) = ran (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)))
10 foeq3 6803 . . . . . 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 586 . . 3 (𝐹 ∈ Fin β†’ dom (𝐹 β†Ύ V) β‰Ό (𝐹 β†Ύ V))
15 resss 6006 . . . 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 584 . 2 (𝐹 ∈ Fin β†’ dom (𝐹 β†Ύ V) β‰Ό 𝐹)
201, 19eqbrtrrid 5184 1 (𝐹 ∈ Fin β†’ dom 𝐹 β‰Ό 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   β†Ύ cres 5678  Rel wrel 5681   Fn wfn 6538  β€“ontoβ†’wfo 6541  β€˜cfv 6543  1st c1st 7972   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 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1st 7974  df-2nd 7975  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-fin 8942
This theorem is referenced by:  dmfi  9329  hashfun  14396
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