Theorem fnfi 7250

Description: A version of fnex 5937 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 5492	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	adantr 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹)
3		reseq2 5058	. . . 4 ⊢ (𝑤 = ∅ → (𝐹 ↾ 𝑤) = (𝐹 ↾ ∅))
4	3	eleq1d 2307	. . 3 ⊢ (𝑤 = ∅ → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
5		reseq2 5058	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
6	5	eleq1d 2307	. . 3 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin))
7		reseq2 5058	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2307	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
9		reseq2 5058	. . . 4 ⊢ (𝑤 = 𝐴 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝐴))
10	9	eleq1d 2307	. . 3 ⊢ (𝑤 = 𝐴 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin))
11		res0 5067	. . . . 5 ⊢ (𝐹 ↾ ∅) = ∅
12		0fi 7188	. . . . 5 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2311	. . . 4 ⊢ (𝐹 ↾ ∅) ∈ Fin
14	13	a1i 9	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
15		resundi 5076	. . . . 5 ⊢ (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧}))
16		simp-4l 547	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝐹 Fn 𝐴)
17		simplrr 542	. . . . . . . . 9 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ (𝐴 ∖ 𝑦))
18	17	eldifad 3231	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ 𝐴)
19		fnressn 5901	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹‘𝑧)⟩})
20	16, 18, 19	syl2anc 415	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹‘𝑧)⟩})
21	20	uneq2d 3383	. . . . . 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 2835	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ V)
24		funfvex 5712	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
25	24	funfni 5483	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ V)
26	16, 18, 25	syl2anc 415	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → (𝐹‘𝑧) ∈ V)
27		opexg 4368	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑧) ∈ V) → ⟨𝑧, (𝐹‘𝑧)⟩ ∈ V)
28	23, 26, 27	syl2anc 415	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ⟨𝑧, (𝐹‘𝑧)⟩ ∈ V)
29	17	eldifbd 3232	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
30		opeldmg 4986	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐹‘𝑧) ∈ V) → (⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦) → 𝑧 ∈ dom (𝐹 ↾ 𝑦)))
31	18, 26, 30	syl2anc 415	. . . . . . . . . 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 5084	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝑦) = (𝑦 ∩ dom 𝐹)
33	32	eleq2i 2305	. . . . . . . . . 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 3412	. . . . . . . . . 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 673	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ¬ ⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦))
39		unsnfi 7226	. . . . . . 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 1278	. . . . . 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 2315	. . . . 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 2325	. . . 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 7196	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)
46	2, 45	eqeltrrd 2316	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∖ cdif 3217   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  {csn 3709  ⟨cop 3712  dom cdm 4774   ↾ cres 4776   Fn wfn 5372  ‘cfv 5377  Fincfn 7022

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1o 6687  df-er 6807  df-en 7023  df-fin 7025

This theorem is used by:  fundmfibi  7252  resfnfinfinss  7253  seqf1oglem2  10957  seqf1og  10958  fihashf1rn  11227  fihashfn  11240  wrdfin  11323  xpsfrnel  13665