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Theorem trpredlem1 31425
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 7037 . . 3 (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
2 fr0g 7476 . . . . . 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 5646 . . . . . 6 Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
42, 3syl6eqss 3634 . . . . 5 (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ⊆ 𝐴)
5 fveq2 6148 . . . . . 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 3611 . . . . 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 236 . . . 4 (𝑖 = ∅ → (Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴))
8 nfcv 2761 . . . . . . . . . . 11 𝑎Pred(𝑅, 𝐴, 𝑋)
9 nfcv 2761 . . . . . . . . . . 11 𝑎𝑗
10 nfmpt1 4707 . . . . . . . . . . . . . . 15 𝑎(𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦))
1110, 8nfrdg 7455 . . . . . . . . . . . . . 14 𝑎rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
12 nfcv 2761 . . . . . . . . . . . . . 14 𝑎ω
1311, 12nfres 5358 . . . . . . . . . . . . 13 𝑎(rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
1413, 9nffv 6155 . . . . . . . . . . . 12 𝑎((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
15 nfcv 2761 . . . . . . . . . . . 12 𝑎Pred(𝑅, 𝐴, 𝑒)
1614, 15nfiun 4514 . . . . . . . . . . 11 𝑎 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒)
17 predeq3 5643 . . . . . . . . . . . . . 14 (𝑦 = 𝑒 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑒))
1817cbviunv 4525 . . . . . . . . . . . . 13 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦) = 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒)
1918mpteq2i 4701 . . . . . . . . . . . 12 (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑎 ∈ V ↦ 𝑒𝑎 Pred(𝑅, 𝐴, 𝑒))
20 rdgeq1 7452 . . . . . . . . . . . 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 5350 . . . . . . . . . . . 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 4500 . . . . . . . . . . 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 7478 . . . . . . . . . 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 4527 . . . . . . . . . . 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 5646 . . . . . . . . . . . 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 2922 . . . . . . . . . 10 𝑒 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑒) ⊆ 𝐴
2924, 28syl6eqss 3634 . . . . . . . . 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 7479 . . . . . . . . . . 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 482 . . . . . . . . . 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 3944 . . . . . . . . . 10 ∅ ⊆ 𝐴
3331, 32syl6eqss 3634 . . . . . . . . 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 831 . . . . . . . 8 (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
3534adantr 481 . . . . . . 7 ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ⊆ 𝐴)
36 fveq2 6148 . . . . . . . . 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 3611 . . . . . . . 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 482 . . . . . . 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 247 . . . . . 6 ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
4039rexlimiva 3021 . . . . 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 394 . . 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 6181 . . . 4 𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ∅)
4544, 32syl6eqss 3634 . . 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 176 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 196  wo 383  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  wss 3555  c0 3891   ciun 4485  cmpt 4673  cres 5076  Predcpred 5638  suc csuc 5684  cfv 5847  ωcom 7012  reccrdg 7450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451
This theorem is referenced by:  trpredss  31427  trpredtr  31428  trpredmintr  31429  trpredrec  31436
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