Theorem fnfi 8798

Description: A version of fnex 6982 for finite sets that does not require Replacement. (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 6468	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	adantr 483	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹)
3		reseq2 5850	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 ↾ 𝑥) = (𝐹 ↾ ∅))
4	3	eleq1d 2899	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
5	4	imbi2d 343	. . . 4 ⊢ (𝑥 = ∅ → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)))
6		reseq2 5850	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝑦))
7	6	eleq1d 2899	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin))
8	7	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin)))
9		reseq2 5850	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
10	9	eleq1d 2899	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
11	10	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
12		reseq2 5850	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝐴))
13	12	eleq1d 2899	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin))
14	13	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)))
15		res0 5859	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅
16		0fin 8748	. . . . . 6 ⊢ ∅ ∈ Fin
17	15, 16	eqeltri 2911	. . . . 5 ⊢ (𝐹 ↾ ∅) ∈ Fin
18	17	a1i 11	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
19		resundi 5869	. . . . . . . 8 ⊢ (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧}))
20		snfi 8596	. . . . . . . . . 10 ⊢ {⟨𝑧, (𝐹‘𝑧)⟩} ∈ Fin
21		fnfun 6455	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
22		funressn 6923	. . . . . . . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
25		ssfi 8740	. . . . . . . . . 10 ⊢ (({⟨𝑧, (𝐹‘𝑧)⟩} ∈ Fin ∧ (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩}) → (𝐹 ↾ {𝑧}) ∈ Fin)
26	20, 24, 25	sylancr 589	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ∈ Fin)
27		unfi 8787	. . . . . . . . 9 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 ↾ {𝑧}) ∈ Fin) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
28	26, 27	sylan2 594	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
29	19, 28	eqeltrid 2919	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
30	29	expcom 416	. . . . . 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 8760	. . 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 2916	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∪ cun 3936   ⊆ wss 3938  ∅c0 4293  {csn 4569  ⟨cop 4575   ↾ cres 5559  Fun wfun 6351   Fn wfn 6352  ‘cfv 6357  Fincfn 8511

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-fin 8515

This theorem is referenced by:  fundmfibi  8805  resfnfinfin  8806  unirnffid  8818  mptfi  8825  seqf1olem2  13413  seqf1o  13414  wrdfin  13884  isstruct2  16495  xpsfrnel  16837  cmpcref  31116  carsggect  31578  ptrecube  34894  ftc1anclem3  34971  sstotbnd2  35054  prdstotbnd  35074  ffi  41436  stoweidlem59  42351  fourierdlem42  42441  fourierdlem54  42452