Theorem imafi 8958

Description: Images of finite sets are finite. For a shorter proof using ax-pow 5288, see imafiALT 9112. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5288. (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 5965	. . . . 5 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = (𝐹 “ ∅))
2	1	eleq1d 2823	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 “ 𝑥) ∈ Fin ↔ (𝐹 “ ∅) ∈ Fin))
3	2	imbi2d 341	. . 3 ⊢ (𝑥 = ∅ → ((Fun 𝐹 → (𝐹 “ 𝑥) ∈ Fin) ↔ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)))
4		imaeq2 5965	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
5	4	eleq1d 2823	. . . 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 5965	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 “ 𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2823	. . . 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 5965	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹 “ 𝑥) = (𝐹 “ 𝑋))
11	10	eleq1d 2823	. . . 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 5985	. . . . 5 ⊢ (𝐹 “ ∅) = ∅
14		0fin 8954	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2835	. . . 4 ⊢ (𝐹 “ ∅) ∈ Fin
16	15	a1i 11	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ ∅) ∈ Fin)
17		funfn 6464	. . . . . . . . . 10 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
18		fnsnfv 6847	. . . . . . . . . 10 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑧 ∈ dom 𝐹) → {(𝐹‘𝑧)} = (𝐹 “ {𝑧}))
19	17, 18	sylanb 581	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → {(𝐹‘𝑧)} = (𝐹 “ {𝑧}))
20		snfi 8834	. . . . . . . . 9 ⊢ {(𝐹‘𝑧)} ∈ Fin
21	19, 20	eqeltrrdi 2848	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
22		ndmima 6011	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) = ∅)
23	22, 14	eqeltrdi 2847	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ dom 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
24	23	adantl 482	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ¬ 𝑧 ∈ dom 𝐹) → (𝐹 “ {𝑧}) ∈ Fin)
25	21, 24	pm2.61dan 810	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 “ {𝑧}) ∈ Fin)
26		imaundi 6053	. . . . . . . 8 ⊢ (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧}))
27		unfi 8955	. . . . . . . 8 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧})) ∈ Fin)
28	26, 27	eqeltrid 2843	. . . . . . 7 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ (𝐹 “ {𝑧}) ∈ Fin) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
29	25, 28	sylan2 593	. . . . . 6 ⊢ (((𝐹 “ 𝑦) ∈ Fin ∧ Fun 𝐹) → (𝐹 “ (𝑦 ∪ {𝑧})) ∈ Fin)
30	29	expcom 414	. . . . 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 8947	. 2 ⊢ (𝑋 ∈ Fin → (Fun 𝐹 → (𝐹 “ 𝑋) ∈ Fin))
34	33	impcom 408	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∪ cun 3885  ∅c0 4256  {csn 4561  dom cdm 5589   “ cima 5592  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433  Fincfn 8733

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-en 8734  df-fin 8737

This theorem is referenced by:  pwfir  8959  pwfilem  8960  fissuni  9124  fipreima  9125  fsuppcolem  9160  cmpfi  22559  mdegldg  25231  mdegcl  25234  trlsegvdeglem6  28589  fsuppcurry1  31060  fsuppcurry2  31061  elrspunidl  31606  locfinreflem  31790  zarcmplem  31831  sibfof  32307  eulerpartlemgf  32346  fineqvrep  33064  poimirlem30  35807  ftc1anclem7  35856  ftc1anc  35858  elrfirn  40517  sge0f1o  43920