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Theorem tfrlem15 7642
Description: Lemma for transfinite recursion. Without assuming ax-rep 4905, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (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
tfrlem15 (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝑓,𝐹,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem15
StepHypRef Expression
1 tfrlem.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem9a 7636 . . 3 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)
32adantl 467 . 2 ((𝐵 ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) ∈ V)
41tfrlem13 7640 . . . 4 ¬ recs(𝐹) ∈ V
5 simpr 471 . . . . 5 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (recs(𝐹) ↾ 𝐵) ∈ V)
6 resss 5564 . . . . . . . 8 (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹)
76a1i 11 . . . . . . 7 (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹))
81tfrlem6 7632 . . . . . . . . 9 Rel recs(𝐹)
9 resdm 5583 . . . . . . . . 9 (Rel recs(𝐹) → (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹))
108, 9ax-mp 5 . . . . . . . 8 (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹)
11 ssres2 5567 . . . . . . . 8 (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ dom recs(𝐹)) ⊆ (recs(𝐹) ↾ 𝐵))
1210, 11syl5eqssr 3800 . . . . . . 7 (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ⊆ (recs(𝐹) ↾ 𝐵))
137, 12eqssd 3770 . . . . . 6 (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
1413eleq1d 2835 . . . . 5 (dom recs(𝐹) ⊆ 𝐵 → ((recs(𝐹) ↾ 𝐵) ∈ V ↔ recs(𝐹) ∈ V))
155, 14syl5ibcom 235 . . . 4 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ∈ V))
164, 15mtoi 190 . . 3 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → ¬ dom recs(𝐹) ⊆ 𝐵)
171tfrlem8 7634 . . . 4 Ord dom recs(𝐹)
18 eloni 5877 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
1918adantr 466 . . . 4 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → Ord 𝐵)
20 ordtri1 5900 . . . . 5 ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (dom recs(𝐹) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ dom recs(𝐹)))
2120con2bid 343 . . . 4 ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵))
2217, 19, 21sylancr 569 . . 3 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵))
2316, 22mpbird 247 . 2 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → 𝐵 ∈ dom recs(𝐹))
243, 23impbida 796 1 (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  wss 3724  dom cdm 5250  cres 5252  Rel wrel 5255  Ord word 5866  Oncon0 5867   Fn wfn 6027  cfv 6032  recscrecs 7621
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 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  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-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-fv 6040  df-wrecs 7560  df-recs 7622
This theorem is referenced by:  tfrlem16  7643  tfr2b  7646
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