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Theorem tfrlem9 6314
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 4819 . . 3 (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) ↔ ∃𝑧𝐵, 𝑧⟩ ∈ recs(𝐹)))
21ibi 176 . 2 (𝐵 ∈ dom recs(𝐹) → ∃𝑧𝐵, 𝑧⟩ ∈ recs(𝐹))
3 df-recs 6300 . . . . . 6 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
43eleq2i 2244 . . . . 5 (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ⟨𝐵, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))})
5 eluniab 3819 . . . . 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 184 . . . 4 (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
7 fnop 5315 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → 𝐵𝑥)
8 rspe 2526 . . . . . . . . . . . . . . . 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 2288 . . . . . . . . . . . . . . . . 17 (𝑓𝐴 ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
11 elssuni 3835 . . . . . . . . . . . . . . . . . 18 (𝑓𝐴𝑓 𝐴)
129recsfval 6310 . . . . . . . . . . . . . . . . . 18 recs(𝐹) = 𝐴
1311, 12sseqtrrdi 3204 . . . . . . . . . . . . . . . . 17 (𝑓𝐴𝑓 ⊆ recs(𝐹))
1410, 13sylbir 135 . . . . . . . . . . . . . . . 16 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) → 𝑓 ⊆ recs(𝐹))
158, 14syl 14 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → 𝑓 ⊆ recs(𝐹))
16 fveq2 5511 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵 → (𝑓𝑦) = (𝑓𝐵))
17 reseq2 4898 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝐵 → (𝑓𝑦) = (𝑓𝐵))
1817fveq2d 5515 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵 → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝑓𝐵)))
1916, 18eqeq12d 2192 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐵 → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝑓𝐵) = (𝐹‘(𝑓𝐵))))
2019rspcv 2837 . . . . . . . . . . . . . . . . . 18 (𝐵𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (𝑓𝐵) = (𝐹‘(𝑓𝐵))))
21 fndm 5311 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
2221eleq2d 2247 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓𝐵𝑥))
239tfrlem7 6312 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Fun recs(𝐹)
24 funssfv 5537 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓𝐵))
2523, 24mp3an1 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓𝐵))
2625adantrl 478 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (recs(𝐹)‘𝐵) = (𝑓𝐵))
2721eleq1d 2246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 Fn 𝑥 → (dom 𝑓 ∈ On ↔ 𝑥 ∈ On))
28 onelss 4384 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝑓 ∈ On → (𝐵 ∈ dom 𝑓𝐵 ⊆ dom 𝑓))
2927, 28syl6bir 164 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓𝐵 ⊆ dom 𝑓)))
3029imp31 256 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → 𝐵 ⊆ dom 𝑓)
31 fun2ssres 5255 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (recs(𝐹) ↾ 𝐵) = (𝑓𝐵))
3231fveq2d 5515 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓𝐵)))
3323, 32mp3an1 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓𝐵)))
3430, 33sylan2 286 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓𝐵)))
3526, 34eqeq12d 2192 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → ((recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)) ↔ (𝑓𝐵) = (𝐹‘(𝑓𝐵))))
3635exbiri 382 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 ⊆ recs(𝐹) → (((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
3736com3l 81 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 Fn 𝑥𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
3837exp31 364 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
3938com34 83 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 Fn 𝑥 → (𝑥 ∈ On → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝐵 ∈ dom 𝑓 → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4039com24 87 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4122, 40sylbird 170 . . . . . . . . . . . . . . . . . . 19 (𝑓 Fn 𝑥 → (𝐵𝑥 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4241com3l 81 . . . . . . . . . . . . . . . . . 18 (𝐵𝑥 → ((𝑓𝐵) = (𝐹‘(𝑓𝐵)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4320, 42syld 45 . . . . . . . . . . . . . . . . 17 (𝐵𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4443com24 87 . . . . . . . . . . . . . . . 16 (𝐵𝑥 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
4544imp4d 352 . . . . . . . . . . . . . . 15 (𝐵𝑥 → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
4615, 45mpdi 43 . . . . . . . . . . . . . 14 (𝐵𝑥 → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
477, 46syl 14 . . . . . . . . . . . . 13 ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
4847exp4d 369 . . . . . . . . . . . 12 ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
4948ex 115 . . . . . . . . . . 11 (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
5049com4r 86 . . . . . . . . . 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 49 . . . . . . . . 9 (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
5251com3l 81 . . . . . . . 8 (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
5352imp4a 349 . . . . . . 7 (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
5453rexlimdv 2593 . . . . . 6 (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
5554imp 124 . . . . 5 ((⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
5655exlimiv 1598 . . . 4 (∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
576, 56sylbi 121 . . 3 (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
5857exlimiv 1598 . 2 (∃𝑧𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
592, 58syl 14 1 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  wrex 2456  wss 3129  cop 3594   cuni 3807  Oncon0 4360  dom cdm 4623  cres 4625  Fun wfun 5206   Fn wfn 5207  cfv 5212  recscrecs 6299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-recs 6300
This theorem is referenced by:  tfr2a  6316  tfrlemiubacc  6325  tfr1onlemubacc  6341  tfrcllemubacc  6354
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