Theorem bnj1093 35610

Description: Technical lemma for bnj69 35640. 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(((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵)
bnj1093.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1093.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Assertion

Ref	Expression
bnj1093	⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 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

Step	Hyp	Ref	Expression
1		bnj1093.2	. . . . . 6 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
2	1	bnj1095 35412	. . . . 5 ⊢ (𝜓 → ∀𝑖𝜓)
3		bnj1093.3	. . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4	2, 3	bnj1096 35413	. . . 4 ⊢ (𝜒 → ∀𝑖𝜒)
5	4	bnj1350 35455	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → ∀𝑖(𝜃 ∧ 𝜏 ∧ 𝜒))
6		bnj1093.1	. . . . 5 ⊢ ∃𝑗(((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵)
7		impexp 456	. . . . . 6 ⊢ ((((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒) → (𝜑0 → (𝑓‘𝑖) ⊆ 𝐵)))
8	7	exbii 1880	. . . . 5 ⊢ (∃𝑗(((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ∃𝑗((𝜃 ∧ 𝜏 ∧ 𝜒) → (𝜑0 → (𝑓‘𝑖) ⊆ 𝐵)))
9	6, 8	mpbi 233	. . . 4 ⊢ ∃𝑗((𝜃 ∧ 𝜏 ∧ 𝜒) → (𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))
10	9	19.37iv 1980	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → ∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))
11	5, 10	alrimih 1856	. 2 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → ∀𝑖∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))
12	11	bnj721 35388	1 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  ∪ ciun 4950  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  ωcom 7877   ∧ w-bnj17 35317   predc-bnj14 35319

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-ral 3077  df-bnj17 35318

This theorem is used by:  bnj1030  35617