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Theorem tfrlem11 7430
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 5753 . 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 7429 . . . . . . . 8 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
5 fnfun 5948 . . . . . . . 8 (𝐶 Fn suc dom recs(𝐹) → Fun 𝐶)
64, 5syl 17 . . . . . . 7 (dom recs(𝐹) ∈ On → Fun 𝐶)
7 ssun1 3759 . . . . . . . . 9 recs(𝐹) ⊆ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
87, 3sseqtr4i 3622 . . . . . . . 8 recs(𝐹) ⊆ 𝐶
92tfrlem9 7427 . . . . . . . . 9 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
10 funssfv 6167 . . . . . . . . . . . 12 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
11103expa 1262 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ∈ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
1211adantrl 751 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐶𝐵) = (recs(𝐹)‘𝐵))
13 onelss 5728 . . . . . . . . . . . 12 (dom recs(𝐹) ∈ On → (𝐵 ∈ dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹)))
1413imp 445 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → 𝐵 ⊆ dom recs(𝐹))
15 fun2ssres 5891 . . . . . . . . . . . . 13 ((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
16153expa 1262 . . . . . . . . . . . 12 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
1716fveq2d 6154 . . . . . . . . . . 11 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ 𝐵 ⊆ dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1814, 17sylan2 491 . . . . . . . . . 10 (((Fun 𝐶 ∧ recs(𝐹) ⊆ 𝐶) ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐹‘(𝐶𝐵)) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
1912, 18eqeq12d 2641 . . . . . . . . 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 716 . . . . . . 7 ((Fun 𝐶 ∧ (dom recs(𝐹) ∈ On ∧ 𝐵 ∈ dom recs(𝐹))) → (𝐵 ∈ dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
226, 21sylan 488 . . . . . 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 4898 . . . . . . . . 9 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ V
2726snid 4184 . . . . . . . 8 𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨𝐵, (𝐹‘(𝐶𝐵))⟩}
28 opeq1 4375 . . . . . . . . . . 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 3641 . . . . . . . . . . . . . 14 (𝐵 = dom recs(𝐹) → 𝐵 ⊆ dom recs(𝐹))
318, 15mp3an2 1409 . . . . . . . . . . . . . 14 ((Fun 𝐶𝐵 ⊆ dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
326, 30, 31syl2an 494 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = (recs(𝐹) ↾ 𝐵))
33 reseq2 5355 . . . . . . . . . . . . . . 15 (𝐵 = dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) = (recs(𝐹) ↾ dom recs(𝐹)))
342tfrlem6 7424 . . . . . . . . . . . . . . . 16 Rel recs(𝐹)
35 resdm 5404 . . . . . . . . . . . . . . . 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 2676 . . . . . . . . . . . . . 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 2660 . . . . . . . . . . . 12 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐶𝐵) = recs(𝐹))
4039fveq2d 6154 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → (𝐹‘(𝐶𝐵)) = (𝐹‘recs(𝐹)))
4140opeq2d 4382 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨dom recs(𝐹), (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4229, 41eqtrd 2660 . . . . . . . . 9 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ = ⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩)
4342sneqd 4165 . . . . . . . 8 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → {⟨𝐵, (𝐹‘(𝐶𝐵))⟩} = {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
4427, 43syl5eleq 2710 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
45 elun2 3764 . . . . . . 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 2715 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶)
484adantr 481 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐶 Fn suc dom recs(𝐹))
49 simpr 477 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 = dom recs(𝐹))
50 sucidg 5765 . . . . . . . 8 (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
5150adantr 481 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → dom recs(𝐹) ∈ suc dom recs(𝐹))
5249, 51eqeltrd 2704 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐵 = dom recs(𝐹)) → 𝐵 ∈ suc dom recs(𝐹))
53 fnopfvb 6195 . . . . . 6 ((𝐶 Fn suc dom recs(𝐹) ∧ 𝐵 ∈ suc dom recs(𝐹)) → ((𝐶𝐵) = (𝐹‘(𝐶𝐵)) ↔ ⟨𝐵, (𝐹‘(𝐶𝐵))⟩ ∈ 𝐶))
5448, 52, 53syl2anc 692 . . . . 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 450 . . 3 (dom recs(𝐹) ∈ On → (𝐵 = dom recs(𝐹) → (𝐶𝐵) = (𝐹‘(𝐶𝐵))))
5725, 56jaod 395 . 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 383  wa 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  wrex 2913  cun 3558  wss 3560  {csn 4153  cop 4159  dom cdm 5079  cres 5081  Rel wrel 5084  Oncon0 5685  suc csuc 5687  Fun wfun 5844   Fn wfn 5845  cfv 5850  recscrecs 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858  df-wrecs 7353  df-recs 7414
This theorem is referenced by:  tfrlem12  7431
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