Theorem fnfi 9183

Description: A version of fnex 7221 for finite sets that does not require Replacement or Power Sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
fnfi	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Proof of Theorem fnfi

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

Step	Hyp	Ref	Expression
1		fnresdm 6669	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	adantr 481	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹)
3		reseq2 5976	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 ↾ 𝑥) = (𝐹 ↾ ∅))
4	3	eleq1d 2818	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
5	4	imbi2d 340	. . . 4 ⊢ (𝑥 = ∅ → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)))
6		reseq2 5976	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝑦))
7	6	eleq1d 2818	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin))
8	7	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin)))
9		reseq2 5976	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
10	9	eleq1d 2818	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
11	10	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
12		reseq2 5976	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝐴))
13	12	eleq1d 2818	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin))
14	13	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)))
15		res0 5985	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅
16		0fin 9173	. . . . . 6 ⊢ ∅ ∈ Fin
17	15, 16	eqeltri 2829	. . . . 5 ⊢ (𝐹 ↾ ∅) ∈ Fin
18	17	a1i 11	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
19		resundi 5995	. . . . . . . 8 ⊢ (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧}))
20		snfi 9046	. . . . . . . . . 10 ⊢ {⟨𝑧, (𝐹‘𝑧)⟩} ∈ Fin
21		fnfun 6649	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
22		funressn 7159	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
24	23	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
25		ssfi 9175	. . . . . . . . . 10 ⊢ (({⟨𝑧, (𝐹‘𝑧)⟩} ∈ Fin ∧ (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩}) → (𝐹 ↾ {𝑧}) ∈ Fin)
26	20, 24, 25	sylancr 587	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ∈ Fin)
27		unfi 9174	. . . . . . . . 9 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 ↾ {𝑧}) ∈ Fin) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
28	26, 27	sylan2 593	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
29	19, 28	eqeltrid 2837	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
30	29	expcom 414	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → ((𝐹 ↾ 𝑦) ∈ Fin → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
31	30	a2i 14	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin) → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
32	31	a1i 11	. . . 4 ⊢ (𝑦 ∈ Fin → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin) → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
33	5, 8, 11, 14, 18, 32	findcard2 9166	. . 3 ⊢ (𝐴 ∈ Fin → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin))
34	33	anabsi7 669	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)
35	2, 34	eqeltrrd 2834	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∪ cun 3946   ⊆ wss 3948  ∅c0 4322  {csn 4628  ⟨cop 4634   ↾ cres 5678  Fun wfun 6537   Fn wfn 6538  ‘cfv 6543  Fincfn 8941

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 7727

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 7858  df-1o 8468  df-en 8942  df-fin 8945

This theorem is referenced by:  f1oenfi  9184  f1oenfirn  9185  f1domfi  9186  f1domfi2  9187  sbthfilem  9203  fundmfibi  9333  resfnfinfin  9334  unirnffid  9346  mptfi  9353  seqf1olem2  14010  seqf1o  14011  wrdfin  14484  isstruct2  17084  xpsfrnel  17510  cmpcref  32899  carsggect  33386  ptrecube  36580  ftc1anclem3  36655  sstotbnd2  36734  prdstotbnd  36754  cantnfub  42159  cantnfub2  42160  ffi  43957  stoweidlem59  44860  fourierdlem42  44950  fourierdlem54  44961