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Theorem bnj518 31811
Description: Technical lemma for bnj852 31846. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj518.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj518.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj518.3 (𝜏 ↔ (𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛))
Assertion
Ref Expression
bnj518 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Distinct variable groups:   𝑓,𝑖,𝑝,𝑦   𝑖,𝑛,𝑝   𝐴,𝑖,𝑝,𝑦   𝑦,𝑅
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛,𝑝)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛,𝑝)   𝜏(𝑥,𝑦,𝑓,𝑖,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑛)   𝑅(𝑥,𝑓,𝑖,𝑛,𝑝)

Proof of Theorem bnj518
StepHypRef Expression
1 bnj518.3 . . . 4 (𝜏 ↔ (𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛))
2 bnj334 31637 . . . 4 ((𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
31, 2bitri 267 . . 3 (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
4 df-bnj17 31611 . . . 4 ((𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛) ↔ ((𝑛 ∈ ω ∧ 𝜑𝜓) ∧ 𝑝𝑛))
5 bnj518.1 . . . . . 6 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
6 bnj518.2 . . . . . 6 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
75, 6bnj517 31810 . . . . 5 ((𝑛 ∈ ω ∧ 𝜑𝜓) → ∀𝑝𝑛 (𝑓𝑝) ⊆ 𝐴)
87r19.21bi 3158 . . . 4 (((𝑛 ∈ ω ∧ 𝜑𝜓) ∧ 𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
94, 8sylbi 209 . . 3 ((𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
103, 9sylbi 209 . 2 (𝜏 → (𝑓𝑝) ⊆ 𝐴)
11 ssel 3852 . . . 4 ((𝑓𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓𝑝) → 𝑦𝐴))
12 bnj93 31788 . . . . 5 ((𝑅 FrSe 𝐴𝑦𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V)
1312ex 405 . . . 4 (𝑅 FrSe 𝐴 → (𝑦𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V))
1411, 13sylan9r 501 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V))
1514ralrimiv 3131 . 2 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
1610, 15sylan2 583 1 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wral 3088  Vcvv 3415  wss 3829  c0 4178   ciun 4792  suc csuc 6031  cfv 6188  ωcom 7396  w-bnj17 31610   predc-bnj14 31612   FrSe w-bnj15 31616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-tr 5031  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fv 6196  df-om 7397  df-bnj17 31611  df-bnj14 31613  df-bnj13 31615  df-bnj15 31617
This theorem is referenced by:  bnj535  31815  bnj546  31821
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