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Theorem imafi 9174
Description: Images of finite sets are finite. For a shorter proof using ax-pow 5363, see imafiALT 9344. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5363. (Revised by BTernaryTau, 7-Sep-2024.)
Assertion
Ref Expression
imafi ((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)

Proof of Theorem imafi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imaeq2 6055 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 “ ∅))
21eleq1d 2818 . . . 4 (𝑥 = ∅ → ((𝐹𝑥) ∈ Fin ↔ (𝐹 “ ∅) ∈ Fin))
32imbi2d 340 . . 3 (𝑥 = ∅ → ((Fun 𝐹 → (𝐹𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)))
4 imaeq2 6055 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
54eleq1d 2818 . . . 4 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝑦) ∈ Fin))
65imbi2d 340 . . 3 (𝑥 = 𝑦 → ((Fun 𝐹 → (𝐹𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹𝑦) ∈ Fin)))
7 imaeq2 6055 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
87eleq1d 2818 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹𝑥) ∈ Fin ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
98imbi2d 340 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((Fun 𝐹 → (𝐹𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)))
10 imaeq2 6055 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1110eleq1d 2818 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝑋) ∈ Fin))
1211imbi2d 340 . . 3 (𝑥 = 𝑋 → ((Fun 𝐹 → (𝐹𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹𝑋) ∈ Fin)))
13 ima0 6076 . . . . 5 (𝐹 “ ∅) = ∅
14 0fin 9170 . . . . 5 ∅ ∈ Fin
1513, 14eqeltri 2829 . . . 4 (𝐹 “ ∅) ∈ Fin
1615a1i 11 . . 3 (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)
17 funfn 6578 . . . . . . . . . 10 (Fun 𝐹𝐹 Fn dom 𝐹)
18 fnsnfv 6970 . . . . . . . . . 10 ((𝐹 Fn dom 𝐹𝑧 ∈ dom 𝐹) → {(𝐹𝑧)} = (𝐹 “ {𝑧}))
1917, 18sylanb 581 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ dom 𝐹) → {(𝐹𝑧)} = (𝐹 “ {𝑧}))
20 snfi 9043 . . . . . . . . 9 {(𝐹𝑧)} ∈ Fin
2119, 20eqeltrrdi 2842 . . . . . . . 8 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
22 ndmima 6102 . . . . . . . . . 10 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) = ∅)
2322, 14eqeltrdi 2841 . . . . . . . . 9 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
2423adantl 482 . . . . . . . 8 ((Fun 𝐹 ∧ ¬ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
2521, 24pm2.61dan 811 . . . . . . 7 (Fun 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
26 imaundi 6149 . . . . . . . 8 (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧}))
27 unfi 9171 . . . . . . . 8 (((𝐹𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → ((𝐹𝑦) ∪ (𝐹 “ {𝑧})) ∈ Fin)
2826, 27eqeltrid 2837 . . . . . . 7 (((𝐹𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
2925, 28sylan2 593 . . . . . 6 (((𝐹𝑦) ∈ Fin ∧ Fun 𝐹) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
3029expcom 414 . . . . 5 (Fun 𝐹 → ((𝐹𝑦) ∈ Fin → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
3130a2i 14 . . . 4 ((Fun 𝐹 → (𝐹𝑦) ∈ Fin) → (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
3231a1i 11 . . 3 (𝑦 ∈ Fin → ((Fun 𝐹 → (𝐹𝑦) ∈ Fin) → (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)))
333, 6, 9, 12, 16, 32findcard2 9163 . 2 (𝑋 ∈ Fin → (Fun 𝐹 → (𝐹𝑋) ∈ Fin))
3433impcom 408 1 ((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  cun 3946  c0 4322  {csn 4628  dom cdm 5676  cima 5679  Fun wfun 6537   Fn wfn 6538  cfv 6543  Fincfn 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-en 8939  df-fin 8942
This theorem is referenced by:  pwfir  9175  pwfilem  9176  fissuni  9356  fipreima  9357  fsuppcolem  9395  cmpfi  22911  mdegldg  25583  mdegcl  25586  trlsegvdeglem6  29475  fsuppcurry1  31945  fsuppcurry2  31946  elrspunidl  32541  locfinreflem  32815  zarcmplem  32856  sibfof  33334  eulerpartlemgf  33373  fineqvrep  34090  poimirlem30  36513  ftc1anclem7  36562  ftc1anc  36564  elrfirn  41423  sge0f1o  45088
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