MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem7 Structured version   Visualization version   GIF version

Theorem tfrlem7 7995
Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem7 Fun recs(𝐹)
Distinct variable group:   𝑥,𝑓,𝑦,𝐹
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem7
Dummy variables 𝑔 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem6 7994 . 2 Rel recs(𝐹)
31recsfval 7993 . . . . . . . . 9 recs(𝐹) = 𝐴
43eleq2i 2902 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑢⟩ ∈ 𝐴)
5 eluni 4815 . . . . . . . 8 (⟨𝑥, 𝑢⟩ ∈ 𝐴 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴))
64, 5bitri 277 . . . . . . 7 (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴))
73eleq2i 2902 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑣⟩ ∈ 𝐴)
8 eluni 4815 . . . . . . . 8 (⟨𝑥, 𝑣⟩ ∈ 𝐴 ↔ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴))
97, 8bitri 277 . . . . . . 7 (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴))
106, 9anbi12i 628 . . . . . 6 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴)))
11 exdistrv 1956 . . . . . 6 (∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ ∃(⟨𝑥, 𝑣⟩ ∈ 𝐴)))
1210, 11bitr4i 280 . . . . 5 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ ∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)))
13 df-br 5041 . . . . . . . . 9 (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
14 df-br 5041 . . . . . . . . 9 (𝑥𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ )
1513, 14anbi12i 628 . . . . . . . 8 ((𝑥𝑔𝑢𝑥𝑣) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ))
161tfrlem5 7992 . . . . . . . . 9 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
1716impcom 410 . . . . . . . 8 (((𝑥𝑔𝑢𝑥𝑣) ∧ (𝑔𝐴𝐴)) → 𝑢 = 𝑣)
1815, 17sylanbr 584 . . . . . . 7 (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ) ∧ (𝑔𝐴𝐴)) → 𝑢 = 𝑣)
1918an4s 658 . . . . . 6 (((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) → 𝑢 = 𝑣)
2019exlimivv 1933 . . . . 5 (∃𝑔((⟨𝑥, 𝑢⟩ ∈ 𝑔𝑔𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ 𝐴)) → 𝑢 = 𝑣)
2112, 20sylbi 219 . . . 4 ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
2221ax-gen 1796 . . 3 𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
2322gen2 1797 . 2 𝑥𝑢𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
24 dffun4 6341 . 2 (Fun recs(𝐹) ↔ (Rel recs(𝐹) ∧ ∀𝑥𝑢𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)))
252, 23, 24mpbir2an 709 1 Fun recs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wal 1535   = wceq 1537  wex 1780  wcel 2114  {cab 2798  wral 3125  wrex 3126  cop 4547   cuni 4812   class class class wbr 5040  cres 5531  Rel wrel 5534  Oncon0 6165  Fun wfun 6323   Fn wfn 6324  cfv 6329  recscrecs 7983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-pss 3930  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4813  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5434  df-eprel 5439  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6122  df-ord 6168  df-on 6169  df-iota 6288  df-fun 6331  df-fn 6332  df-fv 6337  df-wrecs 7923  df-recs 7984
This theorem is referenced by:  tfrlem9  7997  tfrlem9a  7998  tfrlem10  7999  tfrlem14  8003  tfrlem16  8005  tfr1a  8006  tfr1  8009
  Copyright terms: Public domain W3C validator