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Theorem trpredlem1 33151
Description: Technical lemma for transitive predecessors properties. All values of the transitive predecessors' underlying function are subsets of the base set. (Contributed by Scott Fenton, 28-Apr-2012.)
Assertion
Ref Expression
trpredlem1 (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
Distinct variable groups:   𝐴,𝑎,𝑦   𝑅,𝑎,𝑦   𝑋,𝑎
Allowed substitution hints:   𝐴(𝑖)   𝐵(𝑦,𝑖,𝑎)   𝑅(𝑖)   𝑋(𝑦,𝑖)

Proof of Theorem trpredlem1
Dummy variables 𝑒 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0suc 7602 . . 3 (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
2 fr0g 8069 . . . . . 6 (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
3 predss 6144 . . . . . 6 Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
42, 3eqsstrdi 4007 . . . . 5 (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ⊆ 𝐴)
5 fveq2 6663 . . . . . 6 (𝑖 = ∅ → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅))
65sseq1d 3984 . . . . 5 (𝑖 = ∅ → (((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴 ↔ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ⊆ 𝐴))
74, 6syl5ibr 249 . . . 4 (𝑖 = ∅ → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
8 nfcv 2982 . . . . . . . . . . 11 𝑎Pred(𝑅, 𝐴, 𝑋)
9 nfcv 2982 . . . . . . . . . . 11 𝑎𝑗
10 nfmpt1 5151 . . . . . . . . . . . . . . 15 𝑎(𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦))
1110, 8nfrdg 8048 . . . . . . . . . . . . . 14 𝑎rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
12 nfcv 2982 . . . . . . . . . . . . . 14 𝑎ω
1311, 12nfres 5844 . . . . . . . . . . . . 13 𝑎(rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
1413, 9nffv 6673 . . . . . . . . . . . 12 𝑎((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
15 nfcv 2982 . . . . . . . . . . . 12 𝑎Pred(𝑅, 𝐴, 𝑒)
1614, 15nfiun 4935 . . . . . . . . . . 11 𝑎 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒)
17 predeq3 6141 . . . . . . . . . . . . . 14 (𝑦 = 𝑒 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑒))
1817cbviunv 4951 . . . . . . . . . . . . 13 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦) = 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒)
1918mpteq2i 5145 . . . . . . . . . . . 12 (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒))
20 rdgeq1 8045 . . . . . . . . . . . 12 ((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒)) → rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒)), Pred(𝑅, 𝐴, 𝑋)))
21 reseq1 5836 . . . . . . . . . . . 12 (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒)), Pred(𝑅, 𝐴, 𝑋)) → (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω))
2219, 20, 21mp2b 10 . . . . . . . . . . 11 (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
23 iuneq1 4921 . . . . . . . . . . 11 (𝑎 = ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒) = 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒))
248, 9, 16, 22, 23frsucmpt 8071 . . . . . . . . . 10 ((𝑗 ∈ ω ∧ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒))
25 iunss 4955 . . . . . . . . . . 11 ( 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴 ↔ ∀𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴)
26 predss 6144 . . . . . . . . . . . 12 Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴
2726a1i 11 . . . . . . . . . . 11 (𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴)
2825, 27mprgbir 3148 . . . . . . . . . 10 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴
2924, 28eqsstrdi 4007 . . . . . . . . 9 ((𝑗 ∈ ω ∧ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
308, 9, 16, 22, 23frsucmptn 8072 . . . . . . . . . . 11 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = ∅)
3130adantl 485 . . . . . . . . . 10 ((𝑗 ∈ ω ∧ ¬ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = ∅)
32 0ss 4333 . . . . . . . . . 10 ∅ ⊆ 𝐴
3331, 32eqsstrdi 4007 . . . . . . . . 9 ((𝑗 ∈ ω ∧ ¬ 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ∈ V) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
3429, 33pm2.61dan 812 . . . . . . . 8 (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
3534adantr 484 . . . . . . 7 ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
36 fveq2 6663 . . . . . . . . 9 (𝑖 = suc 𝑗 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗))
3736sseq1d 3984 . . . . . . . 8 (𝑖 = suc 𝑗 → (((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴 ↔ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴))
3837adantl 485 . . . . . . 7 ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) → (((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴 ↔ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴))
3935, 38mpbird 260 . . . . . 6 ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
4039rexlimiva 3273 . . . . 5 (∃𝑗 ∈ ω 𝑖 = suc 𝑗 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
4140a1d 25 . . . 4 (∃𝑗 ∈ ω 𝑖 = suc 𝑗 → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
427, 41jaoi 854 . . 3 ((𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗) → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
431, 42syl 17 . 2 (𝑖 ∈ ω → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
44 nfvres 6699 . . . 4 𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ∅)
4544, 32eqsstrdi 4007 . . 3 𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
4645a1d 25 . 2 𝑖 ∈ ω → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
4743, 46pm2.61i 185 1 (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  wrex 3134  Vcvv 3480  wss 3919  c0 4276   ciun 4905  cmpt 5133  cres 5545  Predcpred 6136  suc csuc 6182  cfv 6345  ωcom 7576  reccrdg 8043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-om 7577  df-wrecs 7945  df-recs 8006  df-rdg 8044
This theorem is referenced by:  trpredss  33153  trpredtr  33154  trpredmintr  33155  trpredrec  33162
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