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Theorem bnj1093 30756
Description: Technical lemma for bnj69 30786. 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 30560 . . . . 5 (𝜓 → ∀𝑖𝜓)
3 bnj1093.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
42, 3bnj1096 30561 . . . 4 (𝜒 → ∀𝑖𝜒)
54bnj1350 30604 . . 3 ((𝜃𝜏𝜒) → ∀𝑖(𝜃𝜏𝜒))
6 bnj1093.1 . . . . 5 𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵)
7 impexp 462 . . . . . 6 ((((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝜃𝜏𝜒) → (𝜑0 → (𝑓𝑖) ⊆ 𝐵)))
87exbii 1771 . . . . 5 (∃𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵) ↔ ∃𝑗((𝜃𝜏𝜒) → (𝜑0 → (𝑓𝑖) ⊆ 𝐵)))
96, 8mpbi 220 . . . 4 𝑗((𝜃𝜏𝜒) → (𝜑0 → (𝑓𝑖) ⊆ 𝐵))
10919.37iv 1908 . . 3 ((𝜃𝜏𝜒) → ∃𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
115, 10alrimih 1748 . 2 ((𝜃𝜏𝜒) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
1211bnj721 30535 1 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wral 2907  wss 3555   ciun 4485  suc csuc 5684   Fn wfn 5842  cfv 5847  ωcom 7012  w-bnj17 30459   predc-bnj14 30461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-ral 2912  df-bnj17 30460
This theorem is referenced by:  bnj1030  30763
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