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Theorem fnfi 9218
Description: A version of fnex 7237 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 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)

Proof of Theorem fnfi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnresdm 6687 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
21adantr 480 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) = 𝐹)
3 reseq2 5992 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 ↾ ∅))
43eleq1d 2826 . . . . 5 (𝑥 = ∅ → ((𝐹𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
54imbi2d 340 . . . 4 (𝑥 = ∅ → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)))
6 reseq2 5992 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
76eleq1d 2826 . . . . 5 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝑦) ∈ Fin))
87imbi2d 340 . . . 4 (𝑥 = 𝑦 → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑦) ∈ Fin)))
9 reseq2 5992 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
109eleq1d 2826 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹𝑥) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
1110imbi2d 340 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
12 reseq2 5992 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1312eleq1d 2826 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝐴) ∈ Fin))
1413imbi2d 340 . . . 4 (𝑥 = 𝐴 → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin)))
15 res0 6001 . . . . . 6 (𝐹 ↾ ∅) = ∅
16 0fi 9082 . . . . . 6 ∅ ∈ Fin
1715, 16eqeltri 2837 . . . . 5 (𝐹 ↾ ∅) ∈ Fin
1817a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
19 resundi 6011 . . . . . . . 8 (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧}))
20 snfi 9083 . . . . . . . . . 10 {⟨𝑧, (𝐹𝑧)⟩} ∈ Fin
21 fnfun 6668 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → Fun 𝐹)
22 funressn 7179 . . . . . . . . . . . 12 (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
2321, 22syl 17 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
2423adantr 480 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
25 ssfi 9213 . . . . . . . . . 10 (({⟨𝑧, (𝐹𝑧)⟩} ∈ Fin ∧ (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩}) → (𝐹 ↾ {𝑧}) ∈ Fin)
2620, 24, 25sylancr 587 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ∈ Fin)
27 unfi 9211 . . . . . . . . 9 (((𝐹𝑦) ∈ Fin ∧ (𝐹 ↾ {𝑧}) ∈ Fin) → ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
2826, 27sylan2 593 . . . . . . . 8 (((𝐹𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴𝐴 ∈ Fin)) → ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
2919, 28eqeltrid 2845 . . . . . . 7 (((𝐹𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
3029expcom 413 . . . . . 6 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → ((𝐹𝑦) ∈ Fin → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
3130a2i 14 . . . . 5 (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑦) ∈ Fin) → ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
3231a1i 11 . . . 4 (𝑦 ∈ Fin → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑦) ∈ Fin) → ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
335, 8, 11, 14, 18, 32findcard2 9204 . . 3 (𝐴 ∈ Fin → ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin))
3433anabsi7 671 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin)
352, 34eqeltrrd 2842 1 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cun 3949  wss 3951  c0 4333  {csn 4626  cop 4632  cres 5687  Fun wfun 6555   Fn wfn 6556  cfv 6561  Fincfn 8985
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989
This theorem is referenced by:  f1oenfi  9219  f1oenfirn  9220  f1domfi  9221  f1domfi2  9222  sbthfilem  9238  fodomfir  9368  fundmfibi  9376  resfnfinfin  9377  unirnffid  9387  mptfi  9391  seqf1olem2  14083  seqf1o  14084  wrdfin  14570  isstruct2  17186  xpsfrnel  17607  cyclnumvtx  29820  cmpcref  33849  carsggect  34320  ptrecube  37627  ftc1anclem3  37702  sstotbnd2  37781  prdstotbnd  37801  cantnfub  43334  cantnfub2  43335  ffi  45178  stoweidlem59  46074  fourierdlem42  46164  fourierdlem54  46175
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