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Theorem fnfi 8785
Description: A version of fnex 6972 for finite sets that does not require Replacement. (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 6460 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
21adantr 481 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) = 𝐹)
3 reseq2 5842 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 ↾ ∅))
43eleq1d 2897 . . . . 5 (𝑥 = ∅ → ((𝐹𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
54imbi2d 342 . . . 4 (𝑥 = ∅ → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)))
6 reseq2 5842 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
76eleq1d 2897 . . . . 5 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝑦) ∈ Fin))
87imbi2d 342 . . . 4 (𝑥 = 𝑦 → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑦) ∈ Fin)))
9 reseq2 5842 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
109eleq1d 2897 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹𝑥) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
1110imbi2d 342 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
12 reseq2 5842 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1312eleq1d 2897 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝐴) ∈ Fin))
1413imbi2d 342 . . . 4 (𝑥 = 𝐴 → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin)))
15 res0 5851 . . . . . 6 (𝐹 ↾ ∅) = ∅
16 0fin 8735 . . . . . 6 ∅ ∈ Fin
1715, 16eqeltri 2909 . . . . 5 (𝐹 ↾ ∅) ∈ Fin
1817a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
19 resundi 5861 . . . . . . . 8 (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧}))
20 snfi 8583 . . . . . . . . . 10 {⟨𝑧, (𝐹𝑧)⟩} ∈ Fin
21 fnfun 6447 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → Fun 𝐹)
22 funressn 6914 . . . . . . . . . . . 12 (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
2321, 22syl 17 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
2423adantr 481 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
25 ssfi 8727 . . . . . . . . . 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 8774 . . . . . . . . 9 (((𝐹𝑦) ∈ Fin ∧ (𝐹 ↾ {𝑧}) ∈ Fin) → ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
2826, 27sylan2 592 . . . . . . . 8 (((𝐹𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴𝐴 ∈ Fin)) → ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
2919, 28eqeltrid 2917 . . . . . . 7 (((𝐹𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
3029expcom 414 . . . . . 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 8747 . . 3 (𝐴 ∈ Fin → ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin))
3433anabsi7 667 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin)
352, 34eqeltrrd 2914 1 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cun 3933  wss 3935  c0 4290  {csn 4559  cop 4565  cres 5551  Fun wfun 6343   Fn wfn 6344  cfv 6349  Fincfn 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-fin 8502
This theorem is referenced by:  fundmfibi  8792  resfnfinfin  8793  unirnffid  8805  mptfi  8812  seqf1olem2  13400  seqf1o  13401  wrdfin  13872  isstruct2  16483  xpsfrnel  16825  cmpcref  31014  carsggect  31476  ptrecube  34774  ftc1anclem3  34851  sstotbnd2  34935  prdstotbnd  34955  ffi  41309  stoweidlem59  42225  fourierdlem42  42315  fourierdlem54  42326
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