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Theorem fifo 8495
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 4458 . . . . . 6 (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ≠ ∅)
2 intex 4961 . . . . . 6 (𝑦 ≠ ∅ ↔ 𝑦 ∈ V)
31, 2sylib 208 . . . . 5 (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ∈ V)
43rgen 3052 . . . 4 𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) 𝑦 ∈ V
5 fifo.1 . . . . 5 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑦)
65fnmpt 6173 . . . 4 (∀𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) 𝑦 ∈ V → 𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
74, 6mp1i 13 . . 3 (𝐴𝑉𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
8 dffn4 6274 . . 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 8477 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦))
11 vex 3335 . . . . . 6 𝑥 ∈ V
125elrnmpt 5519 . . . . . 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 2750 . . 3 (𝐴𝑉 → (fi‘𝐴) = ran 𝐹)
16 foeq3 6266 . . 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 1624  wcel 2131  wne 2924  wral 3042  wrex 3043  Vcvv 3332  cdif 3704  cin 3706  c0 4050  𝒫 cpw 4294  {csn 4313   cint 4619  cmpt 4873  ran crn 5259   Fn wfn 6036  ontowfo 6039  cfv 6041  Fincfn 8113  ficfi 8473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-int 4620  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fn 6044  df-fo 6047  df-fv 6049  df-fi 8474
This theorem is referenced by:  inffien  9068  fictb  9251
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