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Theorem tfrlem16 7642
Description: Lemma for finite recursion. Without assuming ax-rep 4904, 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 7633 . . 3 Ord dom recs(𝐹)
3 ordzsl 7192 . . 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 5538 . . . . . . 7 (recs(𝐹) ↾ ∅) = ∅
6 0ex 4924 . . . . . . 7 ∅ ∈ V
75, 6eqeltri 2846 . . . . . 6 (recs(𝐹) ↾ ∅) ∈ V
8 0elon 5921 . . . . . . 7 ∅ ∈ On
91tfrlem15 7641 . . . . . . 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 4070 . . . 4 ¬ dom recs(𝐹) = ∅
1312pm2.21i 117 . . 3 (dom recs(𝐹) = ∅ → Lim dom recs(𝐹))
141tfrlem13 7639 . . . . 5 ¬ recs(𝐹) ∈ V
15 simpr 471 . . . . . . . . . 10 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = suc 𝑧)
16 df-suc 5872 . . . . . . . . . 10 suc 𝑧 = (𝑧 ∪ {𝑧})
1715, 16syl6eq 2821 . . . . . . . . 9 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → dom recs(𝐹) = (𝑧 ∪ {𝑧}))
1817reseq2d 5534 . . . . . . . 8 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → (recs(𝐹) ↾ dom recs(𝐹)) = (recs(𝐹) ↾ (𝑧 ∪ {𝑧})))
191tfrlem6 7631 . . . . . . . . 9 Rel recs(𝐹)
20 resdm 5582 . . . . . . . . 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 5551 . . . . . . . 8 (recs(𝐹) ↾ (𝑧 ∪ {𝑧})) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧}))
2318, 21, 223eqtr3g 2828 . . . . . . 7 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) = ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})))
24 vex 3354 . . . . . . . . . . 11 𝑧 ∈ V
2524sucid 5947 . . . . . . . . . 10 𝑧 ∈ suc 𝑧
2625, 15syl5eleqr 2857 . . . . . . . . 9 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → 𝑧 ∈ dom recs(𝐹))
271tfrlem9a 7635 . . . . . . . . 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 5036 . . . . . . . . 9 {⟨𝑧, (recs(𝐹)‘𝑧)⟩} ∈ V
301tfrlem7 7632 . . . . . . . . . 10 Fun recs(𝐹)
31 funressn 6569 . . . . . . . . . 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 4937 . . . . . . . 8 (recs(𝐹) ↾ {𝑧}) ∈ V
34 unexg 7106 . . . . . . . 8 (((recs(𝐹) ↾ 𝑧) ∈ V ∧ (recs(𝐹) ↾ {𝑧}) ∈ V) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
3528, 33, 34sylancl 574 . . . . . . 7 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → ((recs(𝐹) ↾ 𝑧) ∪ (recs(𝐹) ↾ {𝑧})) ∈ V)
3623, 35eqeltrd 2850 . . . . . 6 ((𝑧 ∈ On ∧ dom recs(𝐹) = suc 𝑧) → recs(𝐹) ∈ V)
3736rexlimiva 3176 . . . . 5 (∃𝑧 ∈ On dom recs(𝐹) = suc 𝑧 → recs(𝐹) ∈ V)
3814, 37mto 188 . . . 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 1539 . 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 382  w3o 1070   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  cun 3721  wss 3723  c0 4063  {csn 4316  cop 4322  dom cdm 5249  cres 5251  Rel wrel 5254  Ord word 5865  Oncon0 5866  Lim wlim 5867  suc csuc 5868  Fun wfun 6025   Fn wfn 6026  cfv 6031  recscrecs 7620
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 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-wrecs 7559  df-recs 7621
This theorem is referenced by:  tfr1a  7643
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