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Theorem fifo 8491
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 4454 . . . . . 6 (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ≠ ∅)
2 intex 4957 . . . . . 6 (𝑦 ≠ ∅ ↔ 𝑦 ∈ V)
31, 2sylib 208 . . . . 5 (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ∈ V)
43rgen 3048 . . . 4 𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) 𝑦 ∈ V
5 fifo.1 . . . . 5 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑦)
65fnmpt 6169 . . . 4 (∀𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) 𝑦 ∈ V → 𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
74, 6mp1i 13 . . 3 (𝐴𝑉𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
8 dffn4 6270 . . 3 (𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)
97, 8sylib 208 . 2 (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)
10 elfi2 8473 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦))
11 vex 3331 . . . . . 6 𝑥 ∈ V
125elrnmpt 5515 . . . . . 6 (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦))
1311, 12ax-mp 5 . . . . 5 (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦)
1410, 13syl6bbr 278 . . . 4 (𝐴𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ 𝑥 ∈ ran 𝐹))
1514eqrdv 2746 . . 3 (𝐴𝑉 → (fi‘𝐴) = ran 𝐹)
16 foeq3 6262 . . 3 ((fi‘𝐴) = ran 𝐹 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹))
1715, 16syl 17 . 2 (𝐴𝑉 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹))
189, 17mpbird 247 1 (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1620  wcel 2127  wne 2920  wral 3038  wrex 3039  Vcvv 3328  cdif 3700  cin 3702  c0 4046  𝒫 cpw 4290  {csn 4309   cint 4615  cmpt 4869  ran crn 5255   Fn wfn 6032  ontowfo 6035  cfv 6037  Fincfn 8109  ficfi 8469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-int 4616  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-fo 6043  df-fv 6045  df-fi 8470
This theorem is referenced by:  inffien  9047  fictb  9230
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