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Theorem tfrlem16 7376
 Description: Lemma for finite recursion. Without assuming ax-rep 4699, 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 7367 . . 3 Ord dom recs(𝐹)
3 ordzsl 6937 . . 3 (Ord dom recs(𝐹) ↔ (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹)))
42, 3mpbi 219 . 2 (dom recs(𝐹) = ∅ ∨ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 ∨ Lim dom recs(𝐹))
5 res0 5321 . . . . . . 7 (recs(𝐹) ↾ ∅) = ∅
6 0ex 4718 . . . . . . 7 ∅ ∈ V
75, 6eqeltri 2684 . . . . . 6 (recs(𝐹) ↾ ∅) ∈ V
8 0elon 5695 . . . . . . 7 ∅ ∈ On
91tfrlem15 7375 . . . . . . 7 (∅ ∈ On → (∅ ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ ∅) ∈ V))
108, 9ax-mp 5 . . . . . 6 (∅ ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ ∅) ∈ V)
117, 10mpbir 220 . . . . 5 ∅ ∈ dom recs(𝐹)
1211n0ii 3881 . . . 4 ¬ dom recs(𝐹) = ∅
1312pm2.21i 115 . . 3 (dom recs(𝐹) = ∅ → Lim dom recs(𝐹))
141tfrlem13 7373 . . . . 5 ¬ recs(𝐹) ∈ V
15 simpr 476 . . . . . . . . . 10 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = suc 𝑧)
16 df-suc 5646 . . . . . . . . . 10 suc 𝑧 = (𝑧 ∪ {𝑧})
1715, 16syl6eq 2660 . . . . . . . . 9 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = (𝑧 ∪ {𝑧}))
1817reseq2d 5317 . . . . . . . 8 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ dom recs(𝐹)) = (recs(𝐹) ↾ (𝑧 ∪ {𝑧})))
191tfrlem6 7365 . . . . . . . . 9 Rel recs(𝐹)
20 resdm 5361 . . . . . . . . 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 5330 . . . . . . . 8 (recs(𝐹) ↾ (𝑧 ∪ {𝑧})) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧}))
2318, 21, 223eqtr3g 2667 . . . . . . 7 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})))
24 vex 3176 . . . . . . . . . . 11 𝑧 ∈ V
2524sucid 5721 . . . . . . . . . 10 𝑧 ∈ suc 𝑧
2625, 15syl5eleqr 2695 . . . . . . . . 9 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → 𝑧 ∈ dom recs(𝐹))
271tfrlem9a 7369 . . . . . . . . 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 4835 . . . . . . . . 9 {⟨𝑧, (recs(𝐹)‘𝑧)⟩} ∈ V
301tfrlem7 7366 . . . . . . . . . 10 Fun recs(𝐹)
31 funressn 6331 . . . . . . . . . 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 4731 . . . . . . . 8 (recs(𝐹) ↾ {𝑧}) ∈ V
34 unexg 6857 . . . . . . . 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 2688 . . . . . 6 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) ∈ V)
3736rexlimiva 3010 . . . . 5 (∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 → recs(𝐹) ∈ V)
3814, 37mto 187 . . . 4 ¬ ∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧
3938pm2.21i 115 . . 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 1383 . 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 195   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  dom cdm 5038   ↾ cres 5040  Rel wrel 5043  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  recscrecs 7354 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-wrecs 7294  df-recs 7355 This theorem is referenced by:  tfr1a  7377
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