Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem9a Structured version   Visualization version   GIF version

Theorem tfrlem9a 7527
 Description: Lemma for transfinite recursion. Without using ax-rep 4804, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem9a (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝑓,𝐹,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem9a
Dummy variables 𝑔 𝑧 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . 5 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem7 7524 . . . 4 Fun recs(𝐹)
3 funfvop 6369 . . . 4 ((Fun recs(𝐹) ∧ 𝐵 ∈ dom recs(𝐹)) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
42, 3mpan 706 . . 3 (𝐵 ∈ dom recs(𝐹) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
51recsfval 7522 . . . . 5 recs(𝐹) = 𝐴
65eleq2i 2722 . . . 4 (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹) ↔ ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝐴)
7 eluni 4471 . . . 4 (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝐴 ↔ ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴))
86, 7bitri 264 . . 3 (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴))
94, 8sylib 208 . 2 (𝐵 ∈ dom recs(𝐹) → ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴))
10 simprr 811 . . . 4 ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) → 𝑔𝐴)
11 vex 3234 . . . . 5 𝑔 ∈ V
121, 11tfrlem3a 7518 . . . 4 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))))
1310, 12sylib 208 . . 3 ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) → ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))))
142a1i 11 . . . . . . . 8 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → Fun recs(𝐹))
15 simplrr 818 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔𝐴)
16 elssuni 4499 . . . . . . . . . 10 (𝑔𝐴𝑔 𝐴)
1715, 16syl 17 . . . . . . . . 9 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 𝐴)
1817, 5syl6sseqr 3685 . . . . . . . 8 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ⊆ recs(𝐹))
19 fndm 6028 . . . . . . . . . . . 12 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
2019ad2antll 765 . . . . . . . . . . 11 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 = 𝑧)
21 simprl 809 . . . . . . . . . . 11 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑧 ∈ On)
2220, 21eqeltrd 2730 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 ∈ On)
23 eloni 5771 . . . . . . . . . 10 (dom 𝑔 ∈ On → Ord dom 𝑔)
2422, 23syl 17 . . . . . . . . 9 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → Ord dom 𝑔)
25 simpll 805 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom recs(𝐹))
26 fvexd 6241 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹)‘𝐵) ∈ V)
27 simplrl 817 . . . . . . . . . . 11 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔)
28 df-br 4686 . . . . . . . . . . 11 (𝐵𝑔(recs(𝐹)‘𝐵) ↔ ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔)
2927, 28sylibr 224 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵𝑔(recs(𝐹)‘𝐵))
30 breldmg 5362 . . . . . . . . . 10 ((𝐵 ∈ dom recs(𝐹) ∧ (recs(𝐹)‘𝐵) ∈ V ∧ 𝐵𝑔(recs(𝐹)‘𝐵)) → 𝐵 ∈ dom 𝑔)
3125, 26, 29, 30syl3anc 1366 . . . . . . . . 9 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom 𝑔)
32 ordelss 5777 . . . . . . . . 9 ((Ord dom 𝑔𝐵 ∈ dom 𝑔) → 𝐵 ⊆ dom 𝑔)
3324, 31, 32syl2anc 694 . . . . . . . 8 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ⊆ dom 𝑔)
34 fun2ssres 5969 . . . . . . . 8 ((Fun recs(𝐹) ∧ 𝑔 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑔) → (recs(𝐹) ↾ 𝐵) = (𝑔𝐵))
3514, 18, 33, 34syl3anc 1366 . . . . . . 7 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹) ↾ 𝐵) = (𝑔𝐵))
3611resex 5478 . . . . . . . 8 (𝑔𝐵) ∈ V
3736a1i 11 . . . . . . 7 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (𝑔𝐵) ∈ V)
3835, 37eqeltrd 2730 . . . . . 6 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹) ↾ 𝐵) ∈ V)
3938expr 642 . . . . 5 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ 𝑧 ∈ On) → (𝑔 Fn 𝑧 → (recs(𝐹) ↾ 𝐵) ∈ V))
4039adantrd 483 . . . 4 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ 𝑧 ∈ On) → ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) → (recs(𝐹) ↾ 𝐵) ∈ V))
4140rexlimdva 3060 . . 3 ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) → (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) → (recs(𝐹) ↾ 𝐵) ∈ V))
4213, 41mpd 15 . 2 ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) → (recs(𝐹) ↾ 𝐵) ∈ V)
439, 42exlimddv 1903 1 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  ⟨cop 4216  ∪ cuni 4468   class class class wbr 4685  dom cdm 5143   ↾ cres 5145  Ord word 5760  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  recscrecs 7512 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-wrecs 7452  df-recs 7513 This theorem is referenced by:  tfrlem15  7533  tfrlem16  7534  rdgseg  7563
 Copyright terms: Public domain W3C validator