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Theorem tfrlem16 7772
Description: Lemma for finite recursion. Without assuming ax-rep 5006, 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 7763 . . 3 Ord dom recs(𝐹)
3 ordzsl 7323 . . 3 (Ord dom recs(𝐹) ↔ (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹)))
42, 3mpbi 222 . 2 (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹))
5 res0 5646 . . . . . . 7 (recs(𝐹) ↾ ∅) = ∅
6 0ex 5026 . . . . . . 7 ∅ ∈ V
75, 6eqeltri 2854 . . . . . 6 (recs(𝐹) ↾ ∅) ∈ V
8 0elon 6029 . . . . . . 7 ∅ ∈ On
91tfrlem15 7771 . . . . . . 7 (∅ ∈ On → (∅ ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ ∅) ∈ V))
108, 9ax-mp 5 . . . . . 6 (∅ ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ ∅) ∈ V)
117, 10mpbir 223 . . . . 5 ∅ ∈ dom recs(𝐹)
1211n0ii 4150 . . . 4 ¬ dom recs(𝐹) = ∅
1312pm2.21i 117 . . 3 (dom recs(𝐹) = ∅ → Lim dom recs(𝐹))
141tfrlem13 7769 . . . . 5 ¬ recs(𝐹) ∈ V
15 simpr 479 . . . . . . . . . 10 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = suc 𝑧)
16 df-suc 5982 . . . . . . . . . 10 suc 𝑧 = (𝑧 ∪ {𝑧})
1715, 16syl6eq 2829 . . . . . . . . 9 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = (𝑧 ∪ {𝑧}))
1817reseq2d 5642 . . . . . . . 8 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ dom recs(𝐹)) = (recs(𝐹) ↾ (𝑧 ∪ {𝑧})))
191tfrlem6 7761 . . . . . . . . 9 Rel recs(𝐹)
20 resdm 5691 . . . . . . . . 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 5660 . . . . . . . 8 (recs(𝐹) ↾ (𝑧 ∪ {𝑧})) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧}))
2318, 21, 223eqtr3g 2836 . . . . . . 7 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})))
24 vex 3400 . . . . . . . . . . 11 𝑧 ∈ V
2524sucid 6055 . . . . . . . . . 10 𝑧 ∈ suc 𝑧
2625, 15syl5eleqr 2865 . . . . . . . . 9 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → 𝑧 ∈ dom recs(𝐹))
271tfrlem9a 7765 . . . . . . . . 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 5140 . . . . . . . . 9 {⟨𝑧, (recs(𝐹)‘𝑧)⟩} ∈ V
301tfrlem7 7762 . . . . . . . . . 10 Fun recs(𝐹)
31 funressn 6692 . . . . . . . . . 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 5040 . . . . . . . 8 (recs(𝐹) ↾ {𝑧}) ∈ V
34 unexg 7236 . . . . . . . 8 (((recs(𝐹) ↾ 𝑧) ∈ V ∧ (recs(𝐹) ↾ {𝑧}) ∈ V) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
3528, 33, 34sylancl 580 . . . . . . 7 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
3623, 35eqeltrd 2858 . . . . . 6 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) ∈ V)
3736rexlimiva 3209 . . . . 5 (∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 → recs(𝐹) ∈ V)
3814, 37mto 189 . . . 4 ¬ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧
3938pm2.21i 117 . . 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 1501 . 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 198  wa 386  w3o 1070   = wceq 1601  wcel 2106  {cab 2762  wral 3089  wrex 3090  Vcvv 3397  cun 3789  wss 3791  c0 4140  {csn 4397  cop 4403  dom cdm 5355  cres 5357  Rel wrel 5360  Ord word 5975  Oncon0 5976  Lim wlim 5977  suc csuc 5978  Fun wfun 6129   Fn wfn 6130  cfv 6135  recscrecs 7750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-wrecs 7689  df-recs 7751
This theorem is referenced by:  tfr1a  7773
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