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Theorem fnfi 8123
 Description: A version of fnex 6386 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 5914 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
21adantr 480 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) = 𝐹)
3 reseq2 5312 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 ↾ ∅))
43eleq1d 2672 . . . . 5 (𝑥 = ∅ → ((𝐹𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
54imbi2d 329 . . . 4 (𝑥 = ∅ → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)))
6 reseq2 5312 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
76eleq1d 2672 . . . . 5 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝑦) ∈ Fin))
87imbi2d 329 . . . 4 (𝑥 = 𝑦 → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑦) ∈ Fin)))
9 reseq2 5312 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
109eleq1d 2672 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹𝑥) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
1110imbi2d 329 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
12 reseq2 5312 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1312eleq1d 2672 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝐴) ∈ Fin))
1413imbi2d 329 . . . 4 (𝑥 = 𝐴 → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin)))
15 res0 5321 . . . . . 6 (𝐹 ↾ ∅) = ∅
16 0fin 8073 . . . . . 6 ∅ ∈ Fin
1715, 16eqeltri 2684 . . . . 5 (𝐹 ↾ ∅) ∈ Fin
1817a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
19 resundi 5330 . . . . . . . 8 (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧}))
20 snfi 7923 . . . . . . . . . 10 {⟨𝑧, (𝐹𝑧)⟩} ∈ Fin
21 fnfun 5902 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → Fun 𝐹)
22 funressn 6331 . . . . . . . . . . . 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 8065 . . . . . . . . . 10 (({⟨𝑧, (𝐹𝑧)⟩} ∈ Fin ∧ (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩}) → (𝐹 ↾ {𝑧}) ∈ Fin)
2620, 24, 25sylancr 694 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ∈ Fin)
27 unfi 8112 . . . . . . . . 9 (((𝐹𝑦) ∈ Fin ∧ (𝐹 ↾ {𝑧}) ∈ Fin) → ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
2826, 27sylan2 490 . . . . . . . 8 (((𝐹𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴𝐴 ∈ Fin)) → ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
2919, 28syl5eqel 2692 . . . . . . 7 (((𝐹𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
3029expcom 450 . . . . . 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 8085 . . 3 (𝐴 ∈ Fin → ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin))
3433anabsi7 856 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin)
352, 34eqeltrrd 2689 1 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  Fincfn 7841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845 This theorem is referenced by:  fundmfibi  8130  unirnffid  8141  mptfi  8148  seqf1olem2  12703  seqf1o  12704  wrdfin  13178  isstruct2  15704  xpsfrnel  16046  usgrafilem2  25941  cmpcref  29245  carsggect  29707  ptrecube  32579  ftc1anclem3  32657  sstotbnd2  32743  prdstotbnd  32763  ffi  38349  stoweidlem59  38952  fourierdlem42  39042  fourierdlem54  39053  resfnfinfin  40339
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