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Theorem tfrlem9 8327
Description: Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem9 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝑓,𝐹,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem9
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eldm2g 5853 . . 3 (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) ↔ ∃𝑧𝐵, 𝑧⟩ ∈ recs(𝐹)))
21ibi 266 . 2 (𝐵 ∈ dom recs(𝐹) → ∃𝑧𝐵, 𝑧⟩ ∈ recs(𝐹))
3 dfrecs3 8314 . . . . . 6 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
43eleq2i 2829 . . . . 5 (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ⟨𝐵, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))})
5 eluniab 4878 . . . . 5 (⟨𝐵, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))} ↔ ∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
64, 5bitri 274 . . . 4 (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
7 fnop 6608 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → 𝐵𝑥)
8 rspe 3230 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
9 tfrlem.1 . . . . . . . . . . . . . . . . . 18 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
109abeq2i 2874 . . . . . . . . . . . . . . . . 17 (𝑓𝐴 ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
11 elssuni 4896 . . . . . . . . . . . . . . . . . 18 (𝑓𝐴𝑓 𝐴)
129recsfval 8323 . . . . . . . . . . . . . . . . . 18 recs(𝐹) = 𝐴
1311, 12sseqtrrdi 3993 . . . . . . . . . . . . . . . . 17 (𝑓𝐴𝑓 ⊆ recs(𝐹))
1410, 13sylbir 234 . . . . . . . . . . . . . . . 16 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) → 𝑓 ⊆ recs(𝐹))
158, 14syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → 𝑓 ⊆ recs(𝐹))
16 fveq2 6839 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵 → (𝑓𝑦) = (𝑓𝐵))
17 reseq2 5930 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝐵 → (𝑓𝑦) = (𝑓𝐵))
1817fveq2d 6843 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵 → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝐵)))
1916, 18eqeq12d 2752 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐵 → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝑓𝐵) = (𝐹‘(𝑓𝐵))))
2019rspcv 3575 . . . . . . . . . . . . . . . . . 18 (𝐵𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (𝑓𝐵) = (𝐹‘(𝑓𝐵))))
21 fndm 6602 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
2221eleq2d 2823 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓𝐵𝑥))
239tfrlem7 8325 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Fun recs(𝐹)
24 funssfv 6860 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓𝐵))
2523, 24mp3an1 1448 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓𝐵))
2625adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (recs(𝐹)‘𝐵) = (𝑓𝐵))
2721eleq1d 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 Fn 𝑥 → (dom 𝑓 ∈ On ↔ 𝑥 ∈ On))
28 onelss 6357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝑓 ∈ On → (𝐵 ∈ dom 𝑓𝐵 ⊆ dom 𝑓))
2927, 28syl6bir 253 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓𝐵 ⊆ dom 𝑓)))
3029imp31 418 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → 𝐵 ⊆ dom 𝑓)
31 fun2ssres 6543 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (recs(𝐹) ↾ 𝐵) = (𝑓𝐵))
3231fveq2d 6843 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓𝐵)))
3323, 32mp3an1 1448 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓𝐵)))
3430, 33sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓𝐵)))
3526, 34eqeq12d 2752 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → ((recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)) ↔ (𝑓𝐵) = (𝐹‘(𝑓𝐵))))
3635exbiri 809 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 ⊆ recs(𝐹) → (((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
3736com3l 89 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
3837exp31 420 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
3938com34 91 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 Fn 𝑥 → (𝑥 ∈ On → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝐵 ∈ dom 𝑓 → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4039com24 95 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4122, 40sylbird 259 . . . . . . . . . . . . . . . . . . 19 (𝑓 Fn 𝑥 → (𝐵𝑥 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4241com3l 89 . . . . . . . . . . . . . . . . . 18 (𝐵𝑥 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4320, 42syld 47 . . . . . . . . . . . . . . . . 17 (𝐵𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4443com24 95 . . . . . . . . . . . . . . . 16 (𝐵𝑥 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4544imp4d 425 . . . . . . . . . . . . . . 15 (𝐵𝑥 → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
4615, 45mpdi 45 . . . . . . . . . . . . . 14 (𝐵𝑥 → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
477, 46syl 17 . . . . . . . . . . . . 13 ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
4847exp4d 434 . . . . . . . . . . . 12 ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
4948ex 413 . . . . . . . . . . 11 (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
5049com4r 94 . . . . . . . . . 10 (𝑓 Fn 𝑥 → (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
5150pm2.43i 52 . . . . . . . . 9 (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
5251com3l 89 . . . . . . . 8 (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
5352imp4a 423 . . . . . . 7 (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
5453rexlimdv 3148 . . . . . 6 (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
5554imp 407 . . . . 5 ((⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
5655exlimiv 1933 . . . 4 (∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
576, 56sylbi 216 . . 3 (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
5857exlimiv 1933 . 2 (∃𝑧𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
592, 58syl 17 1 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3062  wrex 3071  wss 3908  cop 4590   cuni 4863  dom cdm 5631  cres 5633  Oncon0 6315  Fun wfun 6487   Fn wfn 6488  cfv 6493  recscrecs 8312
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-ov 7356  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313
This theorem is referenced by:  tfrlem11  8330  tfr2a  8337
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