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Theorem imafi 9120
Description: Images of finite sets are finite. For a shorter proof using ax-pow 5321, see imafiALT 9290. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5321. (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 6010 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 “ ∅))
21eleq1d 2823 . . . 4 (𝑥 = ∅ → ((𝐹𝑥) ∈ Fin ↔ (𝐹 “ ∅) ∈ Fin))
32imbi2d 341 . . 3 (𝑥 = ∅ → ((Fun 𝐹 → (𝐹𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)))
4 imaeq2 6010 . . . . 5 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
54eleq1d 2823 . . . 4 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝑦) ∈ Fin))
65imbi2d 341 . . 3 (𝑥 = 𝑦 → ((Fun 𝐹 → (𝐹𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹𝑦) ∈ Fin)))
7 imaeq2 6010 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
87eleq1d 2823 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹𝑥) ∈ Fin ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
98imbi2d 341 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((Fun 𝐹 → (𝐹𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)))
10 imaeq2 6010 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1110eleq1d 2823 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝑋) ∈ Fin))
1211imbi2d 341 . . 3 (𝑥 = 𝑋 → ((Fun 𝐹 → (𝐹𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹𝑋) ∈ Fin)))
13 ima0 6030 . . . . 5 (𝐹 “ ∅) = ∅
14 0fin 9116 . . . . 5 ∅ ∈ Fin
1513, 14eqeltri 2834 . . . 4 (𝐹 “ ∅) ∈ Fin
1615a1i 11 . . 3 (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)
17 funfn 6532 . . . . . . . . . 10 (Fun 𝐹𝐹 Fn dom 𝐹)
18 fnsnfv 6921 . . . . . . . . . 10 ((𝐹 Fn dom 𝐹𝑧 ∈ dom 𝐹) → {(𝐹𝑧)} = (𝐹 “ {𝑧}))
1917, 18sylanb 582 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ dom 𝐹) → {(𝐹𝑧)} = (𝐹 “ {𝑧}))
20 snfi 8989 . . . . . . . . 9 {(𝐹𝑧)} ∈ Fin
2119, 20eqeltrrdi 2847 . . . . . . . 8 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
22 ndmima 6056 . . . . . . . . . 10 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) = ∅)
2322, 14eqeltrdi 2846 . . . . . . . . 9 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
2423adantl 483 . . . . . . . 8 ((Fun 𝐹 ∧ ¬ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
2521, 24pm2.61dan 812 . . . . . . 7 (Fun 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
26 imaundi 6103 . . . . . . . 8 (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧}))
27 unfi 9117 . . . . . . . 8 (((𝐹𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → ((𝐹𝑦) ∪ (𝐹 “ {𝑧})) ∈ Fin)
2826, 27eqeltrid 2842 . . . . . . 7 (((𝐹𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
2925, 28sylan2 594 . . . . . 6 (((𝐹𝑦) ∈ Fin ∧ Fun 𝐹) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
3029expcom 415 . . . . 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 9109 . 2 (𝑋 ∈ Fin → (Fun 𝐹 → (𝐹𝑋) ∈ Fin))
3433impcom 409 1 ((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  cun 3909  c0 4283  {csn 4587  dom cdm 5634  cima 5637  Fun wfun 6491   Fn wfn 6492  cfv 6497  Fincfn 8884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-en 8885  df-fin 8888
This theorem is referenced by:  pwfir  9121  pwfilem  9122  fissuni  9302  fipreima  9303  fsuppcolem  9338  cmpfi  22762  mdegldg  25434  mdegcl  25437  trlsegvdeglem6  29172  fsuppcurry1  31645  fsuppcurry2  31646  elrspunidl  32206  locfinreflem  32424  zarcmplem  32465  sibfof  32943  eulerpartlemgf  32982  fineqvrep  33699  poimirlem30  36111  ftc1anclem7  36160  ftc1anc  36162  elrfirn  41021  sge0f1o  44630
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