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Theorem tfrlem11 7638
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 5935 . 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 7637 . . . . . . . 8 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
5 fnfun 6129 . . . . . . . 8 (𝐶 Fn suc dom recs(𝐹) → Fun 𝐶)
64, 5syl 17 . . . . . . 7 (dom recs(𝐹) ∈ On → Fun 𝐶)
7 ssun1 3928 . . . . . . . . 9 recs(𝐹) ⊆ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
87, 3sseqtr4i 3788 . . . . . . . 8 recs(𝐹) ⊆ 𝐶
92tfrlem9 7635 . . . . . . . . 9 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
10 funssfv 6351 . . . . . . . . . . . 12 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
11103expa 1111 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
1211adantrl 689 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
13 onelss 5910 . . . . . . . . . . . 12 (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹)))
1413imp 393 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → 𝐵 ⊆ dom recs(𝐹))
15 fun2ssres 6075 . . . . . . . . . . . . 13 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
16153expa 1111 . . . . . . . . . . . 12 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
1716fveq2d 6337 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1814, 17sylan2 574 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1912, 18eqeq12d 2786 . . . . . . . . 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 675 . . . . . . 7 ((Fun 𝐶 ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
226, 21sylan 563 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
2322exp32 407 . . . . 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 5061 . . . . . . . . 9 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ V
2726snid 4348 . . . . . . . 8 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨𝐵, (𝐹‘(𝐶𝐵))⟩}
28 opeq1 4540 . . . . . . . . . . 11 (𝐵 = dom recs(𝐹) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩)
2928adantl 467 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩)
30 eqimss 3807 . . . . . . . . . . . . . 14 (𝐵 = dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹))
318, 15mp3an2 1560 . . . . . . . . . . . . . 14 ((Fun 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
326, 30, 31syl2an 577 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
33 reseq2 5530 . . . . . . . . . . . . . . 15 (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = (recs(𝐹) ↾ dom recs(𝐹)))
342tfrlem6 7632 . . . . . . . . . . . . . . . 16 Rel recs(𝐹)
35 resdm 5583 . . . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . 14 (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
3837adantl 467 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
3932, 38eqtrd 2805 . . . . . . . . . . . 12 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = recs(𝐹))
4039fveq2d 6337 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘recs(𝐹)))
4140opeq2d 4547 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4229, 41eqtrd 2805 . . . . . . . . 9 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4342sneqd 4329 . . . . . . . 8 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → {⟨𝐵, (𝐹‘(𝐶𝐵))⟩} = {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
4427, 43syl5eleq 2856 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
45 elun2 3933 . . . . . . 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 2861 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶)
484adantr 466 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐶 Fn suc dom recs(𝐹))
49 simpr 471 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 = dom recs(𝐹))
50 sucidg 5947 . . . . . . . 8 (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
5150adantr 466 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → dom recs(𝐹) ∈ suc dom recs(𝐹))
5249, 51eqeltrd 2850 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 ∈ suc dom recs(𝐹))
53 fnopfvb 6379 . . . . . 6 ((𝐶 Fn suc dom recs(𝐹) ∧ 𝐵 ∈ suc dom recs(𝐹)) → ((𝐶𝐵) = (𝐹‘(𝐶𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶))
5448, 52, 53syl2anc 567 . . . . 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 397 . . 3 (dom recs(𝐹) ∈ On → (𝐵 = dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
5725, 56jaod 840 . 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  wa 382  wo 828   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  cun 3722  wss 3724  {csn 4317  cop 4323  dom cdm 5250  cres 5252  Rel wrel 5255  Oncon0 5867  suc csuc 5869  Fun wfun 6026   Fn wfn 6027  cfv 6032  recscrecs 7621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-fv 6040  df-wrecs 7560  df-recs 7622
This theorem is referenced by:  tfrlem12  7639
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