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Theorem tfrlem11 8011
Description: Lemma for transfinite recursion. Compute the value of 𝐶. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlem.3 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
Assertion
Ref Expression
tfrlem11 (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝐶,𝑓,𝑥,𝑦   𝑓,𝐹,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem11
StepHypRef Expression
1 elsuci 6235 . 2 (𝐵 ∈ suc dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) ∨ 𝐵 = dom recs(𝐹)))
2 tfrlem.1 . . . . . . . . 9 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
3 tfrlem.3 . . . . . . . . 9 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
42, 3tfrlem10 8010 . . . . . . . 8 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
5 fnfun 6432 . . . . . . . 8 (𝐶 Fn suc dom recs(𝐹) → Fun 𝐶)
64, 5syl 17 . . . . . . 7 (dom recs(𝐹) ∈ On → Fun 𝐶)
7 ssun1 4123 . . . . . . . . 9 recs(𝐹) ⊆ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
87, 3sseqtrri 3979 . . . . . . . 8 recs(𝐹) ⊆ 𝐶
92tfrlem9 8008 . . . . . . . . 9 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
10 funssfv 6673 . . . . . . . . . . . 12 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
11103expa 1115 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
1211adantrl 715 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
13 onelss 6211 . . . . . . . . . . . 12 (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹)))
1413imp 410 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → 𝐵 ⊆ dom recs(𝐹))
15 fun2ssres 6378 . . . . . . . . . . . . 13 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
16153expa 1115 . . . . . . . . . . . 12 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
1716fveq2d 6656 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1814, 17sylan2 595 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1912, 18eqeq12d 2838 . . . . . . . . 9 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → ((𝐶𝐵) = (𝐹‘(𝐶𝐵)) ↔ (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
209, 19syl5ibr 249 . . . . . . . 8 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
218, 20mpanl2 700 . . . . . . 7 ((Fun 𝐶 ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
226, 21sylan 583 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
2322exp32 424 . . . . 5 (dom recs(𝐹) ∈ On → (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))))
2423pm2.43i 52 . . . 4 (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵)))))
2524pm2.43d 53 . . 3 (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
26 opex 5333 . . . . . . . . 9 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ V
2726snid 4575 . . . . . . . 8 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨𝐵, (𝐹‘(𝐶𝐵))⟩}
28 opeq1 4776 . . . . . . . . . . 11 (𝐵 = dom recs(𝐹) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩)
2928adantl 485 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩)
30 eqimss 3998 . . . . . . . . . . . . . 14 (𝐵 = dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹))
318, 15mp3an2 1446 . . . . . . . . . . . . . 14 ((Fun 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
326, 30, 31syl2an 598 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
33 reseq2 5826 . . . . . . . . . . . . . . 15 (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = (recs(𝐹) ↾ dom recs(𝐹)))
342tfrlem6 8005 . . . . . . . . . . . . . . . 16 Rel recs(𝐹)
35 resdm 5875 . . . . . . . . . . . . . . . 16 (Rel recs(𝐹) → (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹))
3634, 35ax-mp 5 . . . . . . . . . . . . . . 15 (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹)
3733, 36syl6eq 2873 . . . . . . . . . . . . . 14 (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
3837adantl 485 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
3932, 38eqtrd 2857 . . . . . . . . . . . 12 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = recs(𝐹))
4039fveq2d 6656 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘recs(𝐹)))
4140opeq2d 4785 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4229, 41eqtrd 2857 . . . . . . . . 9 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4342sneqd 4551 . . . . . . . 8 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → {⟨𝐵, (𝐹‘(𝐶𝐵))⟩} = {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
4427, 43eleqtrid 2920 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
45 elun2 4128 . . . . . . 7 (⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
4644, 45syl 17 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
4746, 3eleqtrrdi 2925 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶)
48 simpr 488 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 = dom recs(𝐹))
49 sucidg 6247 . . . . . . . 8 (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
5049adantr 484 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → dom recs(𝐹) ∈ suc dom recs(𝐹))
5148, 50eqeltrd 2914 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 ∈ suc dom recs(𝐹))
52 fnopfvb 6701 . . . . . 6 ((𝐶 Fn suc dom recs(𝐹) ∧ 𝐵 ∈ suc dom recs(𝐹)) → ((𝐶𝐵) = (𝐹‘(𝐶𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶))
534, 51, 52syl2an2r 684 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ((𝐶𝐵) = (𝐹‘(𝐶𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶))
5447, 53mpbird 260 . . . 4 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (𝐹‘(𝐶𝐵)))
5554ex 416 . . 3 (dom recs(𝐹) ∈ On → (𝐵 = dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
5625, 55jaod 856 . 2 (dom recs(𝐹) ∈ On → ((𝐵 ∈ dom recs(𝐹) ∨ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
571, 56syl5 34 1 (dom recs(𝐹) ∈ On → (𝐵 ∈ suc dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2114  {cab 2800  wral 3130  wrex 3131  cun 3906  wss 3908  {csn 4539  cop 4545  dom cdm 5532  cres 5534  Rel wrel 5537  Oncon0 6169  suc csuc 6171  Fun wfun 6328   Fn wfn 6329  cfv 6334  recscrecs 7994
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-wrecs 7934  df-recs 7995
This theorem is referenced by:  tfrlem12  8012
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