MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnfi Structured version   Visualization version   GIF version

Theorem fnfi 9216
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 6688 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
21adantr 480 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) = 𝐹)
3 reseq2 5995 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 ↾ ∅))
43eleq1d 2824 . . . . 5 (𝑥 = ∅ → ((𝐹𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈ Fin))
54imbi2d 340 . . . 4 (𝑥 = ∅ → (((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)))
6 reseq2 5995 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
76eleq1d 2824 . . . . 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 5995 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
109eleq1d 2824 . . . . 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 5995 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1312eleq1d 2824 . . . . 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 6004 . . . . . 6 (𝐹 ↾ ∅) = ∅
16 0fi 9081 . . . . . 6 ∅ ∈ Fin
1715, 16eqeltri 2835 . . . . 5 (𝐹 ↾ ∅) ∈ Fin
1817a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈ Fin)
19 resundi 6014 . . . . . . . 8 (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 ↾ {𝑧}))
20 snfi 9082 . . . . . . . . . 10 {⟨𝑧, (𝐹𝑧)⟩} ∈ Fin
21 fnfun 6669 . . . . . . . . . . . 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 9212 . . . . . . . . . 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 9210 . . . . . . . . 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 2843 . . . . . . 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 9203 . . 3 (𝐴 ∈ Fin → ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin))
3433anabsi7 671 . 2 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → (𝐹𝐴) ∈ Fin)
352, 34eqeltrrd 2840 1 ((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cun 3961  wss 3963  c0 4339  {csn 4631  cop 4637  cres 5691  Fun wfun 6557   Fn wfn 6558  cfv 6563  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-fin 8988
This theorem is referenced by:  f1oenfi  9217  f1oenfirn  9218  f1domfi  9219  f1domfi2  9220  sbthfilem  9236  fodomfir  9366  fundmfibi  9374  resfnfinfin  9375  unirnffid  9385  mptfi  9389  seqf1olem2  14080  seqf1o  14081  wrdfin  14567  isstruct2  17183  xpsfrnel  17609  cmpcref  33811  carsggect  34300  ptrecube  37607  ftc1anclem3  37682  sstotbnd2  37761  prdstotbnd  37781  cantnfub  43311  cantnfub2  43312  ffi  45116  stoweidlem59  46015  fourierdlem42  46105  fourierdlem54  46116
  Copyright terms: Public domain W3C validator