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Theorem fidomdm 9279
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 6156 . 2 dom (𝐹 β†Ύ V) = dom 𝐹
2 finresfin 9220 . . . 4 (𝐹 ∈ Fin β†’ (𝐹 β†Ύ V) ∈ Fin)
3 fvex 6859 . . . . . . 7 (1st β€˜π‘₯) ∈ V
4 eqid 2733 . . . . . . 7 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) = (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯))
53, 4fnmpti 6648 . . . . . 6 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) Fn (𝐹 β†Ύ V)
6 dffn4 6766 . . . . . 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 5970 . . . . . 6 Rel (𝐹 β†Ύ V)
9 reldm 7980 . . . . . 6 (Rel (𝐹 β†Ύ V) β†’ dom (𝐹 β†Ύ V) = ran (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)))
10 foeq3 6758 . . . . . 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 9275 . . . 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 587 . . 3 (𝐹 ∈ Fin β†’ dom (𝐹 β†Ύ V) β‰Ό (𝐹 β†Ύ V))
15 resss 5966 . . . 4 (𝐹 β†Ύ V) βŠ† 𝐹
16 ssdomg 8946 . . . 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 8953 . . 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 585 . 2 (𝐹 ∈ Fin β†’ dom (𝐹 β†Ύ V) β‰Ό 𝐹)
201, 19eqbrtrrid 5145 1 (𝐹 ∈ Fin β†’ dom 𝐹 β‰Ό 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  Vcvv 3447   βŠ† wss 3914   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5637  ran crn 5638   β†Ύ cres 5639  Rel wrel 5642   Fn wfn 6495  β€“ontoβ†’wfo 6498  β€˜cfv 6500  1st c1st 7923   β‰Ό cdom 8887  Fincfn 8889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-en 8890  df-dom 8891  df-fin 8893
This theorem is referenced by:  dmfi  9280  hashfun  14346
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