MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem15 Structured version   Visualization version   GIF version

Theorem tfrlem15 7771
Description: Lemma for transfinite recursion. Without assuming ax-rep 5006, 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 7765 . . 3 (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V)
32adantl 475 . 2 ((𝐵 ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) ∈ V)
41tfrlem13 7769 . . . 4 ¬ recs(𝐹) ∈ V
5 simpr 479 . . . . 5 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (recs(𝐹) ↾ 𝐵) ∈ V)
6 resss 5671 . . . . . . . 8 (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹)
76a1i 11 . . . . . . 7 (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹))
81tfrlem6 7761 . . . . . . . . 9 Rel recs(𝐹)
9 resdm 5691 . . . . . . . . 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 5674 . . . . . . . 8 (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ dom recs(𝐹)) ⊆ (recs(𝐹) ↾ 𝐵))
1210, 11syl5eqssr 3869 . . . . . . 7 (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ⊆ (recs(𝐹) ↾ 𝐵))
137, 12eqssd 3838 . . . . . 6 (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) = recs(𝐹))
1413eleq1d 2844 . . . . 5 (dom recs(𝐹) ⊆ 𝐵 → ((recs(𝐹) ↾ 𝐵) ∈ V ↔ recs(𝐹) ∈ V))
155, 14syl5ibcom 237 . . . 4 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ∈ V))
164, 15mtoi 191 . . 3 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → ¬ dom recs(𝐹) ⊆ 𝐵)
171tfrlem8 7763 . . . 4 Ord dom recs(𝐹)
18 eloni 5986 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
1918adantr 474 . . . 4 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → Ord 𝐵)
20 ordtri1 6009 . . . . 5 ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (dom recs(𝐹) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ dom recs(𝐹)))
2120con2bid 346 . . . 4 ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵))
2217, 19, 21sylancr 581 . . 3 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵))
2316, 22mpbird 249 . 2 ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → 𝐵 ∈ dom recs(𝐹))
243, 23impbida 791 1 (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  {cab 2763  wral 3090  wrex 3091  Vcvv 3398  wss 3792  dom cdm 5355  cres 5357  Rel wrel 5360  Ord word 5975  Oncon0 5976   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 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  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 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  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-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143  df-wrecs 7689  df-recs 7751
This theorem is referenced by:  tfrlem16  7772  tfr2b  7775
  Copyright terms: Public domain W3C validator