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Theorem bnj518 32579
Description: Technical lemma for bnj852 32614. 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 32404 . . . 4 ((𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
31, 2bitri 278 . . 3 (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
4 df-bnj17 32378 . . . 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 32578 . . . . 5 ((𝑛 ∈ ω ∧ 𝜑𝜓) → ∀𝑝𝑛 (𝑓𝑝) ⊆ 𝐴)
87r19.21bi 3130 . . . 4 (((𝑛 ∈ ω ∧ 𝜑𝜓) ∧ 𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
94, 8sylbi 220 . . 3 ((𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
103, 9sylbi 220 . 2 (𝜏 → (𝑓𝑝) ⊆ 𝐴)
11 ssel 3893 . . . 4 ((𝑓𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓𝑝) → 𝑦𝐴))
12 bnj93 32556 . . . . 5 ((𝑅 FrSe 𝐴𝑦𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V)
1312ex 416 . . . 4 (𝑅 FrSe 𝐴 → (𝑦𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V))
1411, 13sylan9r 512 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V))
1514ralrimiv 3104 . 2 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
1610, 15sylan2 596 1 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  Vcvv 3408  wss 3866  c0 4237   ciun 4904  suc csuc 6215  cfv 6380  ωcom 7644  w-bnj17 32377   predc-bnj14 32379   FrSe w-bnj15 32383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-tr 5162  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fv 6388  df-om 7645  df-bnj17 32378  df-bnj14 32380  df-bnj13 32382  df-bnj15 32384
This theorem is referenced by:  bnj535  32583  bnj546  32589
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