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Theorem fifo 8874
 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 4698 . . . . . 6 (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ≠ ∅)
2 intex 5216 . . . . . 6 (𝑦 ≠ ∅ ↔ 𝑦 ∈ V)
31, 2sylib 220 . . . . 5 (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ∈ V)
43rgen 3135 . . . 4 𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) 𝑦 ∈ V
5 fifo.1 . . . . 5 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑦)
65fnmpt 6464 . . . 4 (∀𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) 𝑦 ∈ V → 𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
74, 6mp1i 13 . . 3 (𝐴𝑉𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
8 dffn4 6572 . . 3 (𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)
97, 8sylib 220 . 2 (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)
10 elfi2 8856 . . . . 5 (𝐴𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦))
115elrnmpt 5804 . . . . . 6 (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦))
1211elv 3478 . . . . 5 (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = 𝑦)
1310, 12syl6bbr 291 . . . 4 (𝐴𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ 𝑥 ∈ ran 𝐹))
1413eqrdv 2818 . . 3 (𝐴𝑉 → (fi‘𝐴) = ran 𝐹)
15 foeq3 6564 . . 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 259 1 (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114   ≠ wne 3006  ∀wral 3125  ∃wrex 3126  Vcvv 3473   ∖ cdif 3910   ∩ cin 3912  ∅c0 4269  𝒫 cpw 4515  {csn 4543  ∩ cint 4852   ↦ cmpt 5122  ran crn 5532   Fn wfn 6326  –onto→wfo 6329  ‘cfv 6331  Fincfn 8487  ficfi 8852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-int 4853  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-iota 6290  df-fun 6333  df-fn 6334  df-fo 6337  df-fv 6339  df-fi 8853 This theorem is referenced by:  inffien  9467  fictb  9645
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