Theorem bnj1093 34519

Description: Technical lemma for bnj69 34549. 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 34320	. . . . 5 ⊢ (𝜓 → ∀𝑖𝜓)
3		bnj1093.3	. . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4	2, 3	bnj1096 34321	. . . 4 ⊢ (𝜒 → ∀𝑖𝜒)
5	4	bnj1350 34364	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → ∀𝑖(𝜃 ∧ 𝜏 ∧ 𝜒))
6		bnj1093.1	. . . . 5 ⊢ ∃𝑗(((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵)
7		impexp 450	. . . . . 6 ⊢ ((((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒) → (𝜑0 → (𝑓‘𝑖) ⊆ 𝐵)))
8	7	exbii 1842	. . . . 5 ⊢ (∃𝑗(((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ∃𝑗((𝜃 ∧ 𝜏 ∧ 𝜒) → (𝜑0 → (𝑓‘𝑖) ⊆ 𝐵)))
9	6, 8	mpbi 229	. . . 4 ⊢ ∃𝑗((𝜃 ∧ 𝜏 ∧ 𝜒) → (𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))
10	9	19.37iv 1944	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → ∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))
11	5, 10	alrimih 1818	. 2 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → ∀𝑖∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))
12	11	bnj721 34296	1 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3055   ⊆ wss 3943  ∪ ciun 4990  suc csuc 6359   Fn wfn 6531  ‘cfv 6536  ωcom 7851   ∧ w-bnj17 34225   predc-bnj14 34227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-ral 3056  df-bnj17 34226

This theorem is referenced by:  bnj1030  34526