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Theorem tfrlem9 7368
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 5242 . . 3 (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) ↔ ∃𝑧𝐵, 𝑧⟩ ∈ recs(𝐹)))
21ibi 255 . 2 (𝐵 ∈ dom recs(𝐹) → ∃𝑧𝐵, 𝑧⟩ ∈ recs(𝐹))
3 dfrecs3 7356 . . . . . 6 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
43eleq2i 2680 . . . . 5 (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ⟨𝐵, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))})
5 eluniab 4383 . . . . 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 263 . . . 4 (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
7 fnop 5908 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → 𝐵𝑥)
8 rspe 2986 . . . . . . . . . . . . . . . 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 2722 . . . . . . . . . . . . . . . . 17 (𝑓𝐴 ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
11 elssuni 4403 . . . . . . . . . . . . . . . . . 18 (𝑓𝐴𝑓 𝐴)
129recsfval 7364 . . . . . . . . . . . . . . . . . 18 recs(𝐹) = 𝐴
1311, 12syl6sseqr 3615 . . . . . . . . . . . . . . . . 17 (𝑓𝐴𝑓 ⊆ recs(𝐹))
1410, 13sylbir 224 . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵 → (𝑓𝑦) = (𝑓𝐵))
17 reseq2 5312 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝐵 → (𝑓𝑦) = (𝑓𝐵))
1817fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵 → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝐵)))
1916, 18eqeq12d 2625 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐵 → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝑓𝐵) = (𝐹‘(𝑓𝐵))))
2019rspcv 3278 . . . . . . . . . . . . . . . . . 18 (𝐵𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (𝑓𝐵) = (𝐹‘(𝑓𝐵))))
21 fndm 5904 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
2221eleq2d 2673 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓𝐵𝑥))
239tfrlem7 7366 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Fun recs(𝐹)
24 funssfv 6119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓𝐵))
2523, 24mp3an1 1403 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓𝐵))
2625adantrl 748 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (recs(𝐹)‘𝐵) = (𝑓𝐵))
2721eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 Fn 𝑥 → (dom 𝑓 ∈ On ↔ 𝑥 ∈ On))
28 onelss 5683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝑓 ∈ On → (𝐵 ∈ dom 𝑓𝐵 ⊆ dom 𝑓))
2927, 28syl6bir 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓𝐵 ⊆ dom 𝑓)))
3029imp31 447 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → 𝐵 ⊆ dom 𝑓)
31 fun2ssres 5845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (recs(𝐹) ↾ 𝐵) = (𝑓𝐵))
3231fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓𝐵)))
3323, 32mp3an1 1403 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓𝐵)))
3430, 33sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓𝐵)))
3526, 34eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → ((recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)) ↔ (𝑓𝐵) = (𝐹‘(𝑓𝐵))))
3635exbiri 650 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 ⊆ recs(𝐹) → (((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
3736com3l 87 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
3837exp31 628 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
3938com34 89 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 Fn 𝑥 → (𝑥 ∈ On → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝐵 ∈ dom 𝑓 → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4039com24 93 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4122, 40sylbird 249 . . . . . . . . . . . . . . . . . . 19 (𝑓 Fn 𝑥 → (𝐵𝑥 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4241com3l 87 . . . . . . . . . . . . . . . . . 18 (𝐵𝑥 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4320, 42syld 46 . . . . . . . . . . . . . . . . 17 (𝐵𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4443com24 93 . . . . . . . . . . . . . . . 16 (𝐵𝑥 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4544imp4d 616 . . . . . . . . . . . . . . 15 (𝐵𝑥 → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
4615, 45mpdi 44 . . . . . . . . . . . . . 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 635 . . . . . . . . . . . 12 ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
4948ex 449 . . . . . . . . . . 11 (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
5049com4r 92 . . . . . . . . . 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 50 . . . . . . . . 9 (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
5251com3l 87 . . . . . . . 8 (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
5352imp4a 612 . . . . . . 7 (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
5453rexlimdv 3012 . . . . . 6 (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
5554imp 444 . . . . 5 ((⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
5655exlimiv 1845 . . . 4 (∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
576, 56sylbi 206 . . 3 (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
5857exlimiv 1845 . 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 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wral 2896  wrex 2897  wss 3540  cop 4131   cuni 4372  dom cdm 5038  cres 5040  Oncon0 5640  Fun wfun 5798   Fn wfn 5799  cfv 5804  recscrecs 7354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-wrecs 7294  df-recs 7355
This theorem is referenced by:  tfrlem11  7371  tfr2a  7378
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