Theorem fnfi 9102

Description: A version of fnex 7163 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 6611	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	adantr 480	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹)
3		reseq2 5933	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 ↾ 𝑥) = (𝐹 ↾ ∅))
4	3	eleq1d 2821	. . . . 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 5933	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝑦))
7	6	eleq1d 2821	. . . . 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 5933	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
10	9	eleq1d 2821	. . . . 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 5933	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝐴))
13	12	eleq1d 2821	. . . . 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 5942	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅
16		0fi 8979	. . . . . 6 ⊢ ∅ ∈ Fin
17	15, 16	eqeltri 2832	. . . . 5 ⊢ (𝐹 ↾ ∅) ∈ Fin
18	17	a1i 11	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
19		resundi 5952	. . . . . . . 8 ⊢ (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧}))
20		snfi 8980	. . . . . . . . . 10 ⊢ {⟨𝑧, (𝐹‘𝑧)⟩} ∈ Fin
21		fnfun 6592	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
22		funressn 7104	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
25		ssfi 9097	. . . . . . . . . 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 9095	. . . . . . . . 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 2840	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
30	29	expcom 413	. . . . . 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 9089	. . 3 ⊢ (𝐴 ∈ Fin → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin))
34	33	anabsi7 671	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)
35	2, 34	eqeltrrd 2837	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  {csn 4580  ⟨cop 4586   ↾ cres 5626  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-en 8884  df-fin 8887

This theorem is referenced by:  f1oenfi  9103  f1oenfirn  9104  f1domfi  9105  f1domfi2  9106  sbthfilem  9122  fodomfir  9228  fundmfibi  9236  resfnfinfin  9237  unirnffid  9247  mptfi  9251  seqf1olem2  13965  seqf1o  13966  wrdfin  14455  isstruct2  17076  xpsfrnel  17483  cyclnumvtx  29873  cmpcref  34007  carsggect  34475  ptrecube  37821  ftc1anclem3  37896  sstotbnd2  37975  prdstotbnd  37995  cantnfub  43563  cantnfub2  43564  ffi  45417  stoweidlem59  46303  fourierdlem42  46393  fourierdlem54  46404