Theorem imafi 9175

Description: Images of finite sets are finite. For a shorter proof using ax-pow 5364, see imafiALT 9345. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5364. (Revised by BTernaryTau, 7-Sep-2024.)

Assertion

Ref	Expression
imafi	⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)

Proof of Theorem imafi

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaeq2 6056	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = (𝐹 “ ∅))
2	1	eleq1d 2819	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ ∅) ∈ Fin))
3	2	imbi2d 341	. . 3 ⊢ (𝑥 = ∅ → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)))
4		imaeq2 6056	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
5	4	eleq1d 2819	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ 𝑦) ∈ Fin))
6	5	imbi2d 341	. . 3 ⊢ (𝑥 = 𝑦 → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ 𝑦) ∈ Fin)))
7		imaeq2 6056	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 “ 𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2819	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
9	8	imbi2d 341	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)))
10		imaeq2 6056	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹 “ 𝑥) = (𝐹 “ 𝑋))
11	10	eleq1d 2819	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ 𝑋) ∈ Fin))
12	11	imbi2d 341	. . 3 ⊢ (𝑥 = 𝑋 → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ 𝑋) ∈ Fin)))
13		ima0 6077	. . . . 5 ⊢ (𝐹 “ ∅) = ∅
14		0fin 9171	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2830	. . . 4 ⊢ (𝐹 “ ∅) ∈ Fin
16	15	a1i 11	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)
17		funfn 6579	. . . . . . . . . 10 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
18		fnsnfv 6971	. . . . . . . . . 10 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑧 ∈ dom 𝐹) → {(𝐹‘𝑧)} = (𝐹 “ {𝑧}))
19	17, 18	sylanb 582	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → {(𝐹‘𝑧)} = (𝐹 “ {𝑧}))
20		snfi 9044	. . . . . . . . 9 ⊢ {(𝐹‘𝑧)} ∈ Fin
21	19, 20	eqeltrrdi 2843	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
22		ndmima 6103	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) = ∅)
23	22, 14	eqeltrdi 2842	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
24	23	adantl 483	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ¬ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
25	21, 24	pm2.61dan 812	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
26		imaundi 6150	. . . . . . . 8 ⊢ (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧}))
27		unfi 9172	. . . . . . . 8 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧})) ∈ Fin)
28	26, 27	eqeltrid 2838	. . . . . . 7 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
29	25, 28	sylan2 594	. . . . . 6 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ Fun 𝐹) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
30	29	expcom 415	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 “ 𝑦) ∈ Fin → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
31	30	a2i 14	. . . 4 ⊢ ((Fun 𝐹 → (𝐹 “ 𝑦) ∈ Fin) → (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin))
32	31	a1i 11	. . 3 ⊢ (𝑦 ∈ Fin → ((Fun 𝐹 → (𝐹 “ 𝑦) ∈ Fin) → (Fun 𝐹 → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)))
33	3, 6, 9, 12, 16, 32	findcard2 9164	. 2 ⊢ (𝑋 ∈ Fin → (Fun 𝐹 → (𝐹 “ 𝑋) ∈ Fin))
34	33	impcom 409	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∪ cun 3947  ∅c0 4323  {csn 4629  dom cdm 5677   “ cima 5680  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544  Fincfn 8939

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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-en 8940  df-fin 8943

This theorem is referenced by:  pwfir  9176  pwfilem  9177  fissuni  9357  fipreima  9358  fsuppcolem  9396  cmpfi  22912  mdegldg  25584  mdegcl  25587  trlsegvdeglem6  29478  fsuppcurry1  31950  fsuppcurry2  31951  elrspunidl  32546  locfinreflem  32820  zarcmplem  32861  sibfof  33339  eulerpartlemgf  33378  fineqvrep  34095  poimirlem30  36518  ftc1anclem7  36567  ftc1anc  36569  elrfirn  41433  sge0f1o  45098