Theorem fnfi 7099

Description: A version of fnex 5860 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 5431	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	adantr 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹)
3		reseq2 4999	. . . 4 ⊢ (𝑤 = ∅ → (𝐹 ↾ 𝑤) = (𝐹 ↾ ∅))
4	3	eleq1d 2298	. . 3 ⊢ (𝑤 = ∅ → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
5		reseq2 4999	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
6	5	eleq1d 2298	. . 3 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin))
7		reseq2 4999	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
8	7	eleq1d 2298	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
9		reseq2 4999	. . . 4 ⊢ (𝑤 = 𝐴 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝐴))
10	9	eleq1d 2298	. . 3 ⊢ (𝑤 = 𝐴 → ((𝐹 ↾ 𝑤) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin))
11		res0 5008	. . . . 5 ⊢ (𝐹 ↾ ∅) = ∅
12		0fin 7042	. . . . 5 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2302	. . . 4 ⊢ (𝐹 ↾ ∅) ∈ Fin
14	13	a1i 9	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
15		resundi 5017	. . . . 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 3208	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ 𝐴)
19		fnressn 5824	. . . . . . . 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 3358	. . . . . 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 2813	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → 𝑧 ∈ V)
24		funfvex 5643	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
25	24	funfni 5422	. . . . . . . . 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 4313	. . . . . . . 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 3209	. . . . . . . 8 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
30		opeldmg 4927	. . . . . . . . . . 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 5025	. . . . . . . . . . 11 ⊢ dom (𝐹 ↾ 𝑦) = (𝑦 ∩ dom 𝐹)
33	32	eleq2i 2296	. . . . . . . . . 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 3387	. . . . . . . . . 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 667	. . . . . . 7 ⊢ (((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐹 ↾ 𝑦) ∈ Fin) → ¬ ⟨𝑧, (𝐹‘𝑧)⟩ ∈ (𝐹 ↾ 𝑦))
39		unsnfi 7077	. . . . . . 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 1271	. . . . . 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 2306	. . . . 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 2316	. . . 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 7050	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)
46	2, 45	eqeltrrd 2307	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∖ cdif 3194   ∪ cun 3195   ∩ cin 3196   ⊆ wss 3197  ∅c0 3491  {csn 3666  ⟨cop 3669  dom cdm 4718   ↾ cres 4720   Fn wfn 5312  ‘cfv 5317  Fincfn 6885

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560  df-er 6678  df-en 6886  df-fin 6888

This theorem is referenced by:  fundmfibi  7101  resfnfinfinss  7102  seqf1oglem2  10737  seqf1og  10738  fihashf1rn  11005  fihashfn  11017  wrdfin  11085  xpsfrnel  13372