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Theorem fifo 9473
Description: Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
Hypothesis
Ref Expression
fifo.1 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑦)
Assertion
Ref Expression
fifo (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉
Allowed substitution hint:   𝐹(𝑦)

Proof of Theorem fifo
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldifsni 4789 . . . . . 6 (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ≠ ∅)
2 intex 5343 . . . . . 6 (𝑦 ≠ ∅ ↔ 𝑦 ∈ V)
31, 2sylib 218 . . . . 5 (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ∈ V)
43rgen 3062 . . . 4 𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) 𝑦 ∈ V
5 fifo.1 . . . . 5 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑦)
65fnmpt 6707 . . . 4 (∀𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) 𝑦 ∈ V → 𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
74, 6mp1i 13 . . 3 (𝐴𝑉𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
8 dffn4 6825 . . 3 (𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)
97, 8sylib 218 . 2 (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)
10 elfi2 9455 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦))
115elrnmpt 5968 . . . . . 6 (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦))
1211elv 3484 . . . . 5 (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦)
1310, 12bitr4di 289 . . . 4 (𝐴𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ 𝑥 ∈ ran 𝐹))
1413eqrdv 2734 . . 3 (𝐴𝑉 → (fi‘𝐴) = ran 𝐹)
15 foeq3 6817 . . 3 ((fi‘𝐴) = ran 𝐹 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹))
1614, 15syl 17 . 2 (𝐴𝑉 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹))
179, 16mpbird 257 1 (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  cin 3949  c0 4332  𝒫 cpw 4599  {csn 4625   cint 4945  cmpt 5224  ran crn 5685   Fn wfn 6555  ontowfo 6558  cfv 6560  Fincfn 8986  ficfi 9451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-fo 6566  df-fv 6568  df-fi 9452
This theorem is referenced by:  inffien  10104  fictb  10285
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