Theorem fnfi 9119

Description: A version of fnex 7173 for finite sets that does not require Replacement or Power 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 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnresdm 6619	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	adantr 480	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹)
3		reseq2 5934	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 ↾ 𝑥) = (𝐹 ↾ ∅))
4	3	eleq1d 2813	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
5	4	imbi2d 340	. . . 4 ⊢ (𝑥 = ∅ → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)))
6		reseq2 5934	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝑦))
7	6	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin))
8	7	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin)))
9		reseq2 5934	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
10	9	eleq1d 2813	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
11	10	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
12		reseq2 5934	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝐴))
13	12	eleq1d 2813	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin))
14	13	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)))
15		res0 5943	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅
16		0fi 8990	. . . . . 6 ⊢ ∅ ∈ Fin
17	15, 16	eqeltri 2824	. . . . 5 ⊢ (𝐹 ↾ ∅) ∈ Fin
18	17	a1i 11	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
19		resundi 5953	. . . . . . . 8 ⊢ (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧}))
20		snfi 8991	. . . . . . . . . 10 ⊢ {⟨𝑧, (𝐹‘𝑧)⟩} ∈ Fin
21		fnfun 6600	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
22		funressn 7113	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
24	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
25		ssfi 9114	. . . . . . . . . 10 ⊢ (({⟨𝑧, (𝐹‘𝑧)⟩} ∈ Fin ∧ (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩}) → (𝐹 ↾ {𝑧}) ∈ Fin)
26	20, 24, 25	sylancr 587	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ∈ Fin)
27		unfi 9112	. . . . . . . . 9 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 ↾ {𝑧}) ∈ Fin) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
28	26, 27	sylan2 593	. . . . . . . 8 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
29	19, 28	eqeltrid 2832	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
30	29	expcom 413	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → ((𝐹 ↾ 𝑦) ∈ Fin → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
31	30	a2i 14	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin) → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
32	31	a1i 11	. . . 4 ⊢ (𝑦 ∈ Fin → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin) → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
33	5, 8, 11, 14, 18, 32	findcard2 9105	. . 3 ⊢ (𝐴 ∈ Fin → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin))
34	33	anabsi7 671	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)
35	2, 34	eqeltrrd 2829	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  {csn 4585  ⟨cop 4591   ↾ cres 5633  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499  Fincfn 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-en 8896  df-fin 8899

This theorem is referenced by:  f1oenfi  9120  f1oenfirn  9121  f1domfi  9122  f1domfi2  9123  sbthfilem  9139  fodomfir  9255  fundmfibi  9263  resfnfinfin  9264  unirnffid  9274  mptfi  9278  seqf1olem2  13983  seqf1o  13984  wrdfin  14473  isstruct2  17095  xpsfrnel  17501  cyclnumvtx  29703  cmpcref  33813  carsggect  34282  ptrecube  37587  ftc1anclem3  37662  sstotbnd2  37741  prdstotbnd  37761  cantnfub  43283  cantnfub2  43284  ffi  45140  stoweidlem59  46030  fourierdlem42  46120  fourierdlem54  46131