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Theorem tfrlem11 8219
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 6332 . 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 8218 . . . . . . . 8 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
5 fnfun 6533 . . . . . . . 8 (𝐶 Fn suc dom recs(𝐹) → Fun 𝐶)
64, 5syl 17 . . . . . . 7 (dom recs(𝐹) ∈ On → Fun 𝐶)
7 ssun1 4106 . . . . . . . . 9 recs(𝐹) ⊆ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
87, 3sseqtrri 3958 . . . . . . . 8 recs(𝐹) ⊆ 𝐶
92tfrlem9 8216 . . . . . . . . 9 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
10 funssfv 6795 . . . . . . . . . . . 12 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
11103expa 1117 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
1211adantrl 713 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
13 onelss 6308 . . . . . . . . . . . 12 (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹)))
1413imp 407 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → 𝐵 ⊆ dom recs(𝐹))
15 fun2ssres 6479 . . . . . . . . . . . . 13 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
16153expa 1117 . . . . . . . . . . . 12 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
1716fveq2d 6778 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1814, 17sylan2 593 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1912, 18eqeq12d 2754 . . . . . . . . 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 245 . . . . . . . 8 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
218, 20mpanl2 698 . . . . . . 7 ((Fun 𝐶 ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
226, 21sylan 580 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
2322exp32 421 . . . . 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 5379 . . . . . . . . 9 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ V
2726snid 4597 . . . . . . . 8 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨𝐵, (𝐹‘(𝐶𝐵))⟩}
28 opeq1 4804 . . . . . . . . . . 11 (𝐵 = dom recs(𝐹) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩)
2928adantl 482 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩)
30 eqimss 3977 . . . . . . . . . . . . . 14 (𝐵 = dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹))
318, 15mp3an2 1448 . . . . . . . . . . . . . 14 ((Fun 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
326, 30, 31syl2an 596 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
33 reseq2 5886 . . . . . . . . . . . . . . 15 (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = (recs(𝐹) ↾ dom recs(𝐹)))
342tfrlem6 8213 . . . . . . . . . . . . . . . 16 Rel recs(𝐹)
35 resdm 5936 . . . . . . . . . . . . . . . 16 (Rel recs(𝐹) → (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹))
3634, 35ax-mp 5 . . . . . . . . . . . . . . 15 (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹)
3733, 36eqtrdi 2794 . . . . . . . . . . . . . 14 (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
3837adantl 482 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
3932, 38eqtrd 2778 . . . . . . . . . . . 12 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = recs(𝐹))
4039fveq2d 6778 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘recs(𝐹)))
4140opeq2d 4811 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4229, 41eqtrd 2778 . . . . . . . . 9 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4342sneqd 4573 . . . . . . . 8 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → {⟨𝐵, (𝐹‘(𝐶𝐵))⟩} = {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
4427, 43eleqtrid 2845 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
45 elun2 4111 . . . . . . 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 2850 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶)
48 simpr 485 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 = dom recs(𝐹))
49 sucidg 6344 . . . . . . . 8 (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
5049adantr 481 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → dom recs(𝐹) ∈ suc dom recs(𝐹))
5148, 50eqeltrd 2839 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 ∈ suc dom recs(𝐹))
52 fnopfvb 6823 . . . . . 6 ((𝐶 Fn suc dom recs(𝐹) ∧ 𝐵 ∈ suc dom recs(𝐹)) → ((𝐶𝐵) = (𝐹‘(𝐶𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶))
534, 51, 52syl2an2r 682 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ((𝐶𝐵) = (𝐹‘(𝐶𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶))
5447, 53mpbird 256 . . . 4 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (𝐹‘(𝐶𝐵)))
5554ex 413 . . 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 205  wa 396  wo 844   = wceq 1539  wcel 2106  {cab 2715  wral 3064  wrex 3065  cun 3885  wss 3887  {csn 4561  cop 4567  dom cdm 5589  cres 5591  Rel wrel 5594  Oncon0 6266  suc csuc 6268  Fun wfun 6427   Fn wfn 6428  cfv 6433  recscrecs 8201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202
This theorem is referenced by:  tfrlem12  8220
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