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Theorem tfrlem11 7529
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 5829 . 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 7528 . . . . . . . 8 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
5 fnfun 6026 . . . . . . . 8 (𝐶 Fn suc dom recs(𝐹) → Fun 𝐶)
64, 5syl 17 . . . . . . 7 (dom recs(𝐹) ∈ On → Fun 𝐶)
7 ssun1 3809 . . . . . . . . 9 recs(𝐹) ⊆ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
87, 3sseqtr4i 3671 . . . . . . . 8 recs(𝐹) ⊆ 𝐶
92tfrlem9 7526 . . . . . . . . 9 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
10 funssfv 6247 . . . . . . . . . . . 12 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
11103expa 1284 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
1211adantrl 752 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
13 onelss 5804 . . . . . . . . . . . 12 (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹)))
1413imp 444 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → 𝐵 ⊆ dom recs(𝐹))
15 fun2ssres 5969 . . . . . . . . . . . . 13 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
16153expa 1284 . . . . . . . . . . . 12 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
1716fveq2d 6233 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1814, 17sylan2 490 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1912, 18eqeq12d 2666 . . . . . . . . 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 236 . . . . . . . 8 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
218, 20mpanl2 717 . . . . . . 7 ((Fun 𝐶 ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
226, 21sylan 487 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
2322exp32 630 . . . . 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 4962 . . . . . . . . 9 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ V
2726snid 4241 . . . . . . . 8 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨𝐵, (𝐹‘(𝐶𝐵))⟩}
28 opeq1 4433 . . . . . . . . . . 11 (𝐵 = dom recs(𝐹) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩)
2928adantl 481 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩)
30 eqimss 3690 . . . . . . . . . . . . . 14 (𝐵 = dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹))
318, 15mp3an2 1452 . . . . . . . . . . . . . 14 ((Fun 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
326, 30, 31syl2an 493 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
33 reseq2 5423 . . . . . . . . . . . . . . 15 (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = (recs(𝐹) ↾ dom recs(𝐹)))
342tfrlem6 7523 . . . . . . . . . . . . . . . 16 Rel recs(𝐹)
35 resdm 5476 . . . . . . . . . . . . . . . 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 2701 . . . . . . . . . . . . . 14 (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
3837adantl 481 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
3932, 38eqtrd 2685 . . . . . . . . . . . 12 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = recs(𝐹))
4039fveq2d 6233 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘recs(𝐹)))
4140opeq2d 4440 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4229, 41eqtrd 2685 . . . . . . . . 9 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4342sneqd 4222 . . . . . . . 8 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → {⟨𝐵, (𝐹‘(𝐶𝐵))⟩} = {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
4427, 43syl5eleq 2736 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
45 elun2 3814 . . . . . . 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, 3syl6eleqr 2741 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶)
484adantr 480 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐶 Fn suc dom recs(𝐹))
49 simpr 476 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 = dom recs(𝐹))
50 sucidg 5841 . . . . . . . 8 (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
5150adantr 480 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → dom recs(𝐹) ∈ suc dom recs(𝐹))
5249, 51eqeltrd 2730 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 ∈ suc dom recs(𝐹))
53 fnopfvb 6275 . . . . . 6 ((𝐶 Fn suc dom recs(𝐹) ∧ 𝐵 ∈ suc dom recs(𝐹)) → ((𝐶𝐵) = (𝐹‘(𝐶𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶))
5448, 52, 53syl2anc 694 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ((𝐶𝐵) = (𝐹‘(𝐶𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶))
5547, 54mpbird 247 . . . 4 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (𝐹‘(𝐶𝐵)))
5655ex 449 . . 3 (dom recs(𝐹) ∈ On → (𝐵 = dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
5725, 56jaod 394 . 2 (dom recs(𝐹) ∈ On → ((𝐵 ∈ dom recs(𝐹) ∨ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
581, 57syl5 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 196  wo 382  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  cun 3605  wss 3607  {csn 4210  cop 4216  dom cdm 5143  cres 5145  Rel wrel 5148  Oncon0 5761  suc csuc 5763  Fun wfun 5920   Fn wfn 5921  cfv 5926  recscrecs 7512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-wrecs 7452  df-recs 7513
This theorem is referenced by:  tfrlem12  7530
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