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Theorem fnfi 8189
 Description: A version of fnex 6441 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 5963 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
21adantr 481 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) = 𝐹)
3 reseq2 5356 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 ↾ ∅))
43eleq1d 2683 . . . . 5 (𝑥 = ∅ → ((𝐹𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
54imbi2d 330 . . . 4 (𝑥 = ∅ → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)))
6 reseq2 5356 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
76eleq1d 2683 . . . . 5 (𝑥 = 𝑦 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝑦) ∈ Fin))
87imbi2d 330 . . . 4 (𝑥 = 𝑦 → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑦) ∈ Fin)))
9 reseq2 5356 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
109eleq1d 2683 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹𝑥) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))
1110imbi2d 330 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)))
12 reseq2 5356 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1312eleq1d 2683 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ Fin ↔ (𝐹𝐴) ∈ Fin))
1413imbi2d 330 . . . 4 (𝑥 = 𝐴 → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin)))
15 res0 5365 . . . . . 6 (𝐹 ↾ ∅) = ∅
16 0fin 8139 . . . . . 6 ∅ ∈ Fin
1715, 16eqeltri 2694 . . . . 5 (𝐹 ↾ ∅) ∈ Fin
1817a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
19 resundi 5374 . . . . . . . 8 (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧}))
20 snfi 7989 . . . . . . . . . 10 {⟨𝑧, (𝐹𝑧)⟩} ∈ Fin
21 fnfun 5951 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 → Fun 𝐹)
22 funressn 6386 . . . . . . . . . . . 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 8131 . . . . . . . . . 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 8178 . . . . . . . . 9 (((𝐹𝑦) ∈ Fin ∧ (𝐹 ↾ {𝑧}) ∈ Fin) → ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
2826, 27sylan2 491 . . . . . . . 8 (((𝐹𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴𝐴 ∈ Fin)) → ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin)
2919, 28syl5eqel 2702 . . . . . . 7 (((𝐹𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)
3029expcom 451 . . . . . 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 8151 . . 3 (𝐴 ∈ Fin → ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin))
3433anabsi7 859 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin)
352, 34eqeltrrd 2699 1 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ∪ cun 3557   ⊆ wss 3559  ∅c0 3896  {csn 4153  ⟨cop 4159   ↾ cres 5081  Fun wfun 5846   Fn wfn 5847  ‘cfv 5852  Fincfn 7906 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7907  df-fin 7910 This theorem is referenced by:  fundmfibi  8196  resfnfinfin  8197  unirnffid  8209  mptfi  8216  seqf1olem2  12788  seqf1o  12789  wrdfin  13269  isstruct2  15797  xpsfrnel  16151  cmpcref  29717  carsggect  30179  ptrecube  33068  ftc1anclem3  33146  sstotbnd2  33232  prdstotbnd  33252  ffi  38851  stoweidlem59  39604  fourierdlem42  39694  fourierdlem54  39705
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