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Theorem tfrlem16 7435
Description: Lemma for finite recursion. Without assuming ax-rep 4736, we can show that the domain of the constructed function is a limit ordinal, and hence contains all the finite ordinals. (Contributed by Mario Carneiro, 14-Nov-2014.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem16 Lim dom recs(𝐹)
Distinct variable group:   𝑥,𝑓,𝑦,𝐹
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem16
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem8 7426 . . 3 Ord dom recs(𝐹)
3 ordzsl 6993 . . 3 (Ord dom recs(𝐹) ↔ (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹)))
42, 3mpbi 220 . 2 (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹))
5 res0 5364 . . . . . . 7 (recs(𝐹) ↾ ∅) = ∅
6 0ex 4755 . . . . . . 7 ∅ ∈ V
75, 6eqeltri 2700 . . . . . 6 (recs(𝐹) ↾ ∅) ∈ V
8 0elon 5740 . . . . . . 7 ∅ ∈ On
91tfrlem15 7434 . . . . . . 7 (∅ ∈ On → (∅ ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ ∅) ∈ V))
108, 9ax-mp 5 . . . . . 6 (∅ ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ ∅) ∈ V)
117, 10mpbir 221 . . . . 5 ∅ ∈ dom recs(𝐹)
1211n0ii 3903 . . . 4 ¬ dom recs(𝐹) = ∅
1312pm2.21i 116 . . 3 (dom recs(𝐹) = ∅ → Lim dom recs(𝐹))
141tfrlem13 7432 . . . . 5 ¬ recs(𝐹) ∈ V
15 simpr 477 . . . . . . . . . 10 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = suc 𝑧)
16 df-suc 5691 . . . . . . . . . 10 suc 𝑧 = (𝑧 ∪ {𝑧})
1715, 16syl6eq 2676 . . . . . . . . 9 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = (𝑧 ∪ {𝑧}))
1817reseq2d 5360 . . . . . . . 8 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ dom recs(𝐹)) = (recs(𝐹) ↾ (𝑧 ∪ {𝑧})))
191tfrlem6 7424 . . . . . . . . 9 Rel recs(𝐹)
20 resdm 5404 . . . . . . . . 9 (Rel recs(𝐹) → (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹))
2119, 20ax-mp 5 . . . . . . . 8 (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹)
22 resundi 5373 . . . . . . . 8 (recs(𝐹) ↾ (𝑧 ∪ {𝑧})) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧}))
2318, 21, 223eqtr3g 2683 . . . . . . 7 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})))
24 vex 3194 . . . . . . . . . . 11 𝑧 ∈ V
2524sucid 5766 . . . . . . . . . 10 𝑧 ∈ suc 𝑧
2625, 15syl5eleqr 2711 . . . . . . . . 9 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → 𝑧 ∈ dom recs(𝐹))
271tfrlem9a 7428 . . . . . . . . 9 (𝑧 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝑧) ∈ V)
2826, 27syl 17 . . . . . . . 8 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ 𝑧) ∈ V)
29 snex 4874 . . . . . . . . 9 {⟨𝑧, (recs(𝐹)‘𝑧)⟩} ∈ V
301tfrlem7 7425 . . . . . . . . . 10 Fun recs(𝐹)
31 funressn 6381 . . . . . . . . . 10 (Fun recs(𝐹) → (recs(𝐹) ↾ {𝑧}) ⊆ {⟨𝑧, (recs(𝐹)‘𝑧)⟩})
3230, 31ax-mp 5 . . . . . . . . 9 (recs(𝐹) ↾ {𝑧}) ⊆ {⟨𝑧, (recs(𝐹)‘𝑧)⟩}
3329, 32ssexi 4768 . . . . . . . 8 (recs(𝐹) ↾ {𝑧}) ∈ V
34 unexg 6913 . . . . . . . 8 (((recs(𝐹) ↾ 𝑧) ∈ V ∧ (recs(𝐹) ↾ {𝑧}) ∈ V) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
3528, 33, 34sylancl 693 . . . . . . 7 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
3623, 35eqeltrd 2704 . . . . . 6 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) ∈ V)
3736rexlimiva 3026 . . . . 5 (∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 → recs(𝐹) ∈ V)
3814, 37mto 188 . . . 4 ¬ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧
3938pm2.21i 116 . . 3 (∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 → Lim dom recs(𝐹))
40 id 22 . . 3 (Lim dom recs(𝐹) → Lim dom recs(𝐹))
4113, 39, 403jaoi 1388 . 2 ((dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹)) → Lim dom recs(𝐹))
424, 41ax-mp 5 1 Lim dom recs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3o 1035   = wceq 1480  wcel 1992  {cab 2612  wral 2912  wrex 2913  Vcvv 3191  cun 3558  wss 3560  c0 3896  {csn 4153  cop 4159  dom cdm 5079  cres 5081  Rel wrel 5084  Ord word 5684  Oncon0 5685  Lim wlim 5686  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-reu 2919  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-pw 4137  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-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-wrecs 7353  df-recs 7414
This theorem is referenced by:  tfr1a  7436
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