Theorem fnfi 6902

Description: A version of fnex 5707 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 5297	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	adantr 274	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹)
3		reseq2 4879	. . . 4 ⊢ (𝑤 = ∅ → (𝐹 ↾ 𝑤) = (𝐹 ↾ ∅))
4	3	eleq1d 2235	. . 3 ⊢ (𝑤 = ∅ → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
5		reseq2 4879	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
6	5	eleq1d 2235	. . 3 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin))
7		reseq2 4879	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2235	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
9		reseq2 4879	. . . 4 ⊢ (𝑤 = 𝐴 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝐴))
10	9	eleq1d 2235	. . 3 ⊢ (𝑤 = 𝐴 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin))
11		res0 4888	. . . . 5 ⊢ (𝐹 ↾ ∅) = ∅
12		0fin 6850	. . . . 5 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2239	. . . 4 ⊢ (𝐹 ↾ ∅) ∈ Fin
14	13	a1i 9	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
15		resundi 4897	. . . . 5 ⊢ (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧}))
16		simp-4l 531	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝐹 Fn 𝐴)
17		simplrr 526	. . . . . . . . 9 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ (𝐴 ∖ 𝑦))
18	17	eldifad 3127	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ 𝐴)
19		fnressn 5671	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹‘𝑧)⟩})
20	16, 18, 19	syl2anc 409	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹‘𝑧)⟩})
21	20	uneq2d 3276	. . . . . 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 109	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin)
23	17	elexd 2739	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ V)
24		funfvex 5503	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
25	24	funfni 5288	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ V)
26	16, 18, 25	syl2anc 409	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹‘𝑧) ∈ V)
27		opexg 4206	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑧) ∈ V) → ⟨𝑧, (𝐹‘𝑧)⟩ ∈ V)
28	23, 26, 27	syl2anc 409	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ⟨𝑧, (𝐹‘𝑧)⟩ ∈ V)
29	17	eldifbd 3128	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
30		opeldmg 4809	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ V) → (⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦) → 𝑧 ∈ dom (𝐹 ↾ 𝑦)))
31	18, 26, 30	syl2anc 409	. . . . . . . . . 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 4905	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝑦) = (𝑦 ∩ dom 𝐹)
33	32	eleq2i 2233	. . . . . . . . . 10 ⊢ (𝑧 ∈ dom (𝐹 ↾ 𝑦) ↔ 𝑧 ∈ (𝑦 ∩ dom 𝐹))
34	31, 33	syl6ib 160	. . . . . . . . 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 3305	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑦 ∩ dom 𝐹) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ dom 𝐹))
36	35	simplbi 272	. . . . . . . . 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 653	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ¬ ⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦))
39		unsnfi 6884	. . . . . . 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 1228	. . . . . 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 2243	. . . . 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 2253	. . . 4 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
43	42	ex 114	. . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐹 ↾ 𝑦) ∈ Fin → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
44		simpr 109	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
45	4, 6, 8, 10, 14, 43, 44	findcard2sd 6858	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)
46	2, 45	eqeltrrd 2244	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ∖ cdif 3113   ∪ cun 3114   ∩ cin 3115   ⊆ wss 3116  ∅c0 3409  {csn 3576  ⟨cop 3579  dom cdm 4604   ↾ cres 4606   Fn wfn 5183  ‘cfv 5188  Fincfn 6706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1o 6384  df-er 6501  df-en 6707  df-fin 6709

This theorem is referenced by:  fundmfibi  6904  resfnfinfinss  6905  fihashf1rn  10702  fihashfn  10713