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Theorem fidomdm 9359
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 6207 . 2 dom (𝐹 β†Ύ V) = dom 𝐹
2 finresfin 9299 . . . 4 (𝐹 ∈ Fin β†’ (𝐹 β†Ύ V) ∈ Fin)
3 fvex 6913 . . . . . . 7 (1st β€˜π‘₯) ∈ V
4 eqid 2727 . . . . . . 7 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) = (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯))
53, 4fnmpti 6701 . . . . . 6 (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)) Fn (𝐹 β†Ύ V)
6 dffn4 6820 . . . . . 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 6013 . . . . . 6 Rel (𝐹 β†Ύ V)
9 reldm 8052 . . . . . 6 (Rel (𝐹 β†Ύ V) β†’ dom (𝐹 β†Ύ V) = ran (π‘₯ ∈ (𝐹 β†Ύ V) ↦ (1st β€˜π‘₯)))
10 foeq3 6812 . . . . . 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 9355 . . . 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 584 . . 3 (𝐹 ∈ Fin β†’ dom (𝐹 β†Ύ V) β‰Ό (𝐹 β†Ύ V))
15 resss 6009 . . . 4 (𝐹 β†Ύ V) βŠ† 𝐹
16 ssdomg 9025 . . . 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 9032 . . 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 582 . 2 (𝐹 ∈ Fin β†’ dom (𝐹 β†Ύ V) β‰Ό 𝐹)
201, 19eqbrtrrid 5186 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 3471   βŠ† wss 3947   class class class wbr 5150   ↦ cmpt 5233  dom cdm 5680  ran crn 5681   β†Ύ cres 5682  Rel wrel 5685   Fn wfn 6546  β€“ontoβ†’wfo 6549  β€˜cfv 6551  1st c1st 7995   β‰Ό cdom 8966  Fincfn 8968
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-om 7875  df-1st 7997  df-2nd 7998  df-1o 8491  df-er 8729  df-en 8969  df-dom 8970  df-fin 8972
This theorem is referenced by:  dmfi  9360  hashfun  14434
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