Theorem fnfi 7002

Description: A version of fnex 5784 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 5367	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	adantr 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹)
3		reseq2 4941	. . . 4 ⊢ (𝑤 = ∅ → (𝐹 ↾ 𝑤) = (𝐹 ↾ ∅))
4	3	eleq1d 2265	. . 3 ⊢ (𝑤 = ∅ → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
5		reseq2 4941	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
6	5	eleq1d 2265	. . 3 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin))
7		reseq2 4941	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2265	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
9		reseq2 4941	. . . 4 ⊢ (𝑤 = 𝐴 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝐴))
10	9	eleq1d 2265	. . 3 ⊢ (𝑤 = 𝐴 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin))
11		res0 4950	. . . . 5 ⊢ (𝐹 ↾ ∅) = ∅
12		0fin 6945	. . . . 5 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2269	. . . 4 ⊢ (𝐹 ↾ ∅) ∈ Fin
14	13	a1i 9	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
15		resundi 4959	. . . . 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 3168	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ 𝐴)
19		fnressn 5748	. . . . . . . 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 3317	. . . . . 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 2776	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ V)
24		funfvex 5575	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
25	24	funfni 5358	. . . . . . . . 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 4261	. . . . . . . 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 3169	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
30		opeldmg 4871	. . . . . . . . . . 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 4967	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝑦) = (𝑦 ∩ dom 𝐹)
33	32	eleq2i 2263	. . . . . . . . . 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 3346	. . . . . . . . . 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 6980	. . . . . . 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 2273	. . . . 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 2283	. . . 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 6953	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)
46	2, 45	eqeltrrd 2274	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∖ cdif 3154   ∪ cun 3155   ∩ cin 3156   ⊆ wss 3157  ∅c0 3450  {csn 3622  ⟨cop 3625  dom cdm 4663   ↾ cres 4665   Fn wfn 5253  ‘cfv 5258  Fincfn 6799

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1o 6474  df-er 6592  df-en 6800  df-fin 6802

This theorem is referenced by:  fundmfibi  7004  resfnfinfinss  7005  seqf1oglem2  10612  seqf1og  10613  fihashf1rn  10880  fihashfn  10892  wrdfin  10954  xpsfrnel  12987