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Theorem tfrlem9a 8005
Description: Lemma for transfinite recursion. Without using ax-rep 5154, 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 8002 . . . 4 Fun recs(𝐹)
3 funfvop 6797 . . . 4 ((Fun recs(𝐹) ∧ 𝐵 ∈ dom recs(𝐹)) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
42, 3mpan 689 . . 3 (𝐵 ∈ dom recs(𝐹) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹))
51recsfval 8000 . . . . 5 recs(𝐹) = 𝐴
65eleq2i 2881 . . . 4 (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹) ↔ ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝐴)
7 eluni 4803 . . . 4 (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝐴 ↔ ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴))
86, 7bitri 278 . . 3 (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴))
94, 8sylib 221 . 2 (𝐵 ∈ dom recs(𝐹) → ∃𝑔(⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴))
10 simprr 772 . . . 4 ((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) → 𝑔𝐴)
11 vex 3444 . . . . 5 𝑔 ∈ V
121, 11tfrlem3a 7996 . . . 4 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))))
1310, 12sylib 221 . . 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 777 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔𝐴)
16 elssuni 4830 . . . . . . . . . 10 (𝑔𝐴𝑔 𝐴)
1715, 16syl 17 . . . . . . . . 9 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 𝐴)
1817, 5sseqtrrdi 3966 . . . . . . . 8 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑔 ⊆ recs(𝐹))
19 fndm 6425 . . . . . . . . . . . 12 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
2019ad2antll 728 . . . . . . . . . . 11 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 = 𝑧)
21 simprl 770 . . . . . . . . . . 11 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝑧 ∈ On)
2220, 21eqeltrd 2890 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → dom 𝑔 ∈ On)
23 eloni 6169 . . . . . . . . . 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 766 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom recs(𝐹))
26 fvexd 6660 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹)‘𝐵) ∈ V)
27 simplrl 776 . . . . . . . . . . 11 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔)
28 df-br 5031 . . . . . . . . . . 11 (𝐵𝑔(recs(𝐹)‘𝐵) ↔ ⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔)
2927, 28sylibr 237 . . . . . . . . . 10 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵𝑔(recs(𝐹)‘𝐵))
30 breldmg 5742 . . . . . . . . . 10 ((𝐵 ∈ dom recs(𝐹) ∧ (recs(𝐹)‘𝐵) ∈ V ∧ 𝐵𝑔(recs(𝐹)‘𝐵)) → 𝐵 ∈ dom 𝑔)
3125, 26, 29, 30syl3anc 1368 . . . . . . . . 9 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ∈ dom 𝑔)
32 ordelss 6175 . . . . . . . . 9 ((Ord dom 𝑔𝐵 ∈ dom 𝑔) → 𝐵 ⊆ dom 𝑔)
3324, 31, 32syl2anc 587 . . . . . . . 8 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → 𝐵 ⊆ dom 𝑔)
34 fun2ssres 6369 . . . . . . . 8 ((Fun recs(𝐹) ∧ 𝑔 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑔) → (recs(𝐹) ↾ 𝐵) = (𝑔𝐵))
3514, 18, 33, 34syl3anc 1368 . . . . . . 7 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹) ↾ 𝐵) = (𝑔𝐵))
3611resex 5866 . . . . . . . 8 (𝑔𝐵) ∈ V
3736a1i 11 . . . . . . 7 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (𝑔𝐵) ∈ V)
3835, 37eqeltrd 2890 . . . . . 6 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ (𝑧 ∈ On ∧ 𝑔 Fn 𝑧)) → (recs(𝐹) ↾ 𝐵) ∈ V)
3938expr 460 . . . . 5 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ 𝑧 ∈ On) → (𝑔 Fn 𝑧 → (recs(𝐹) ↾ 𝐵) ∈ V))
4039adantrd 495 . . . 4 (((𝐵 ∈ dom recs(𝐹) ∧ (⟨𝐵, (recs(𝐹)‘𝐵)⟩ ∈ 𝑔𝑔𝐴)) ∧ 𝑧 ∈ On) → ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) → (recs(𝐹) ↾ 𝐵) ∈ V))
4140rexlimdva 3243 . . 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 1936 1 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2111  {cab 2776  wral 3106  wrex 3107  Vcvv 3441  wss 3881  cop 4531   cuni 4800   class class class wbr 5030  dom cdm 5519  cres 5521  Ord word 6158  Oncon0 6159  Fun wfun 6318   Fn wfn 6319  cfv 6324  recscrecs 7990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-wrecs 7930  df-recs 7991
This theorem is referenced by:  tfrlem15  8011  tfrlem16  8012  rdgseg  8041
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