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Theorem bnj518 30210
 Description: Technical lemma for bnj852 30245. 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 30032 . . . 4 ((𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
31, 2bitri 263 . . 3 (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
4 df-bnj17 30006 . . . 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 30209 . . . . 5 ((𝑛 ∈ ω ∧ 𝜑𝜓) → ∀𝑝𝑛 (𝑓𝑝) ⊆ 𝐴)
87r19.21bi 2916 . . . 4 (((𝑛 ∈ ω ∧ 𝜑𝜓) ∧ 𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
94, 8sylbi 206 . . 3 ((𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
103, 9sylbi 206 . 2 (𝜏 → (𝑓𝑝) ⊆ 𝐴)
11 ssel 3562 . . . 4 ((𝑓𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓𝑝) → 𝑦𝐴))
12 bnj93 30187 . . . . 5 ((𝑅 FrSe 𝐴𝑦𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V)
1312ex 449 . . . 4 (𝑅 FrSe 𝐴 → (𝑦𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V))
1411, 13sylan9r 688 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V))
1514ralrimiv 2948 . 2 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
1610, 15sylan2 490 1 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  suc csuc 5642  ‘cfv 5804  ωcom 6957   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fv 5812  df-om 6958  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012 This theorem is referenced by:  bnj535  30214  bnj546  30220
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