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Theorem bnj518 31336
Description: Technical lemma for bnj852 31371. 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 31162 . . . 4 ((𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
31, 2bitri 266 . . 3 (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
4 df-bnj17 31136 . . . 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 31335 . . . . 5 ((𝑛 ∈ ω ∧ 𝜑𝜓) → ∀𝑝𝑛 (𝑓𝑝) ⊆ 𝐴)
87r19.21bi 3079 . . . 4 (((𝑛 ∈ ω ∧ 𝜑𝜓) ∧ 𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
94, 8sylbi 208 . . 3 ((𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
103, 9sylbi 208 . 2 (𝜏 → (𝑓𝑝) ⊆ 𝐴)
11 ssel 3755 . . . 4 ((𝑓𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓𝑝) → 𝑦𝐴))
12 bnj93 31313 . . . . 5 ((𝑅 FrSe 𝐴𝑦𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V)
1312ex 401 . . . 4 (𝑅 FrSe 𝐴 → (𝑦𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V))
1411, 13sylan9r 504 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V))
1514ralrimiv 3112 . 2 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
1610, 15sylan2 586 1 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  wss 3732  c0 4079   ciun 4676  suc csuc 5910  cfv 6068  ωcom 7263  w-bnj17 31135   predc-bnj14 31137   FrSe w-bnj15 31141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fv 6076  df-om 7264  df-bnj17 31136  df-bnj14 31138  df-bnj13 31140  df-bnj15 31142
This theorem is referenced by:  bnj535  31340  bnj546  31346
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