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Theorem bnj1093 32366
 Description: Technical lemma for bnj69 32396. 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
bnj1093.1 𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵)
bnj1093.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1093.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
Assertion
Ref Expression
bnj1093 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
Distinct variable groups:   𝜒,𝑗   𝜏,𝑖   𝜃,𝑖   𝜏,𝑗   𝜃,𝑗   𝐷,𝑖   𝑓,𝑖   𝑖,𝑛   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑓,𝑛)   𝜏(𝑦,𝑓,𝑛)   𝜁(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑗,𝑛)   𝑅(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜑0(𝑦,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1093
StepHypRef Expression
1 bnj1093.2 . . . . . 6 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
21bnj1095 32167 . . . . 5 (𝜓 → ∀𝑖𝜓)
3 bnj1093.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
42, 3bnj1096 32168 . . . 4 (𝜒 → ∀𝑖𝜒)
54bnj1350 32211 . . 3 ((𝜃𝜏𝜒) → ∀𝑖(𝜃𝜏𝜒))
6 bnj1093.1 . . . . 5 𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵)
7 impexp 454 . . . . . 6 ((((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝜃𝜏𝜒) → (𝜑0 → (𝑓𝑖) ⊆ 𝐵)))
87exbii 1849 . . . . 5 (∃𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵) ↔ ∃𝑗((𝜃𝜏𝜒) → (𝜑0 → (𝑓𝑖) ⊆ 𝐵)))
96, 8mpbi 233 . . . 4 𝑗((𝜃𝜏𝜒) → (𝜑0 → (𝑓𝑖) ⊆ 𝐵))
10919.37iv 1949 . . 3 ((𝜃𝜏𝜒) → ∃𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
115, 10alrimih 1825 . 2 ((𝜃𝜏𝜒) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
1211bnj721 32142 1 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2112  ∀wral 3109   ⊆ wss 3884  ∪ ciun 4884  suc csuc 6165   Fn wfn 6323  ‘cfv 6328  ωcom 7564   ∧ w-bnj17 32070   predc-bnj14 32072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-ral 3114  df-bnj17 32071 This theorem is referenced by:  bnj1030  32373
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