Theorem fnfi 6967

Description: A version of fnex 5759 for finite 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 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnresdm 5344	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	adantr 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹)
3		reseq2 4920	. . . 4 ⊢ (𝑤 = ∅ → (𝐹 ↾ 𝑤) = (𝐹 ↾ ∅))
4	3	eleq1d 2258	. . 3 ⊢ (𝑤 = ∅ → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
5		reseq2 4920	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
6	5	eleq1d 2258	. . 3 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin))
7		reseq2 4920	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2258	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
9		reseq2 4920	. . . 4 ⊢ (𝑤 = 𝐴 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝐴))
10	9	eleq1d 2258	. . 3 ⊢ (𝑤 = 𝐴 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin))
11		res0 4929	. . . . 5 ⊢ (𝐹 ↾ ∅) = ∅
12		0fin 6913	. . . . 5 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2262	. . . 4 ⊢ (𝐹 ↾ ∅) ∈ Fin
14	13	a1i 9	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
15		resundi 4938	. . . . 5 ⊢ (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧}))
16		simp-4l 541	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝐹 Fn 𝐴)
17		simplrr 536	. . . . . . . . 9 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ (𝐴 ∖ 𝑦))
18	17	eldifad 3155	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ 𝐴)
19		fnressn 5723	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹‘𝑧)⟩})
20	16, 18, 19	syl2anc 411	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹‘𝑧)⟩})
21	20	uneq2d 3304	. . . . . 6 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ {⟨𝑧, (𝐹‘𝑧)⟩}))
22		simpr 110	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin)
23	17	elexd 2765	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ V)
24		funfvex 5551	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
25	24	funfni 5335	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ V)
26	16, 18, 25	syl2anc 411	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹‘𝑧) ∈ V)
27		opexg 4246	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑧) ∈ V) → ⟨𝑧, (𝐹‘𝑧)⟩ ∈ V)
28	23, 26, 27	syl2anc 411	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ⟨𝑧, (𝐹‘𝑧)⟩ ∈ V)
29	17	eldifbd 3156	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
30		opeldmg 4850	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ V) → (⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦) → 𝑧 ∈ dom (𝐹 ↾ 𝑦)))
31	18, 26, 30	syl2anc 411	. . . . . . . . . 10 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦) → 𝑧 ∈ dom (𝐹 ↾ 𝑦)))
32		dmres 4946	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝑦) = (𝑦 ∩ dom 𝐹)
33	32	eleq2i 2256	. . . . . . . . . 10 ⊢ (𝑧 ∈ dom (𝐹 ↾ 𝑦) ↔ 𝑧 ∈ (𝑦 ∩ dom 𝐹))
34	31, 33	imbitrdi 161	. . . . . . . . 9 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦) → 𝑧 ∈ (𝑦 ∩ dom 𝐹)))
35		elin 3333	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑦 ∩ dom 𝐹) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ dom 𝐹))
36	35	simplbi 274	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑦 ∩ dom 𝐹) → 𝑧 ∈ 𝑦)
37	34, 36	syl6 33	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦) → 𝑧 ∈ 𝑦))
38	29, 37	mtod 664	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ¬ ⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦))
39		unsnfi 6948	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ ⟨𝑧, (𝐹‘𝑧)⟩ ∈ V ∧ ¬ ⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦)) → ((𝐹 ↾ 𝑦) ∪ {⟨𝑧, (𝐹‘𝑧)⟩}) ∈ Fin)
40	22, 28, 38, 39	syl3anc 1249	. . . . . 6 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ((𝐹 ↾ 𝑦) ∪ {⟨𝑧, (𝐹‘𝑧)⟩}) ∈ Fin)
41	21, 40	eqeltrd 2266	. . . . 5 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
42	15, 41	eqeltrid 2276	. . . 4 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
43	42	ex 115	. . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐹 ↾ 𝑦) ∈ Fin → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
44		simpr 110	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
45	4, 6, 8, 10, 14, 43, 44	findcard2sd 6921	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)
46	2, 45	eqeltrrd 2267	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ∖ cdif 3141   ∪ cun 3142   ∩ cin 3143   ⊆ wss 3144  ∅c0 3437  {csn 3607  ⟨cop 3610  dom cdm 4644   ↾ cres 4646   Fn wfn 5230  ‘cfv 5235  Fincfn 6767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1o 6442  df-er 6560  df-en 6768  df-fin 6770

This theorem is referenced by:  fundmfibi  6969  resfnfinfinss  6970  fihashf1rn  10803  fihashfn  10815  xpsfrnel  12823