Theorem bnj539 35588

Description: Technical lemma for bnj852 35618. 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
bnj539.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj539.2	⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓)
bnj539.3	⊢ 𝑀 ∈ V

Assertion

Ref	Expression
bnj539	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑀 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑛   𝑛,𝐹   𝑖,𝑀   𝑅,𝑛   𝑖,𝑛   𝑦,𝑛

Allowed substitution hints:   𝜓(𝑦, 𝑖, 𝑛)   𝐴(𝑦, 𝑖)   𝑅(𝑦, 𝑖)   𝐹(𝑦, 𝑖)   𝑀(𝑦, 𝑛)   𝜓′(𝑦, 𝑖, 𝑛)

Proof of Theorem bnj539

Step	Hyp	Ref	Expression
1		bnj539.2	. 2 ⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓)
2		bnj539.1	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3	2	sbcbii 3798	. . 3 ⊢ ([𝑀 / 𝑛]𝜓 ↔ [𝑀 / 𝑛]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4		bnj539.3	. . . . 5 ⊢ 𝑀 ∈ V
5	4	bnj538 35438	. . . 4 ⊢ ([𝑀 / 𝑛]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω [𝑀 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
6		sbcimg 3790	. . . . . . 7 ⊢ (𝑀 ∈ V → ([𝑀 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝑀 / 𝑛]suc 𝑖 ∈ 𝑛 → [𝑀 / 𝑛](𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))))
7	4, 6	ax-mp 5	. . . . . 6 ⊢ ([𝑀 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝑀 / 𝑛]suc 𝑖 ∈ 𝑛 → [𝑀 / 𝑛](𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
8		sbcel2gv 3808	. . . . . . . 8 ⊢ (𝑀 ∈ V → ([𝑀 / 𝑛]suc 𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ 𝑀))
9	4, 8	ax-mp 5	. . . . . . 7 ⊢ ([𝑀 / 𝑛]suc 𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ 𝑀)
10	4	bnj525 35436	. . . . . . 7 ⊢ ([𝑀 / 𝑛](𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))
11	9, 10	imbi12i 353	. . . . . 6 ⊢ (([𝑀 / 𝑛]suc 𝑖 ∈ 𝑛 → [𝑀 / 𝑛](𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑀 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
12	7, 11	bitri 278	. . . . 5 ⊢ ([𝑀 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑀 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
13	12	ralbii 3112	. . . 4 ⊢ (∀𝑖 ∈ ω [𝑀 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑀 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
14	5, 13	bitri 278	. . 3 ⊢ ([𝑀 / 𝑛]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑀 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
15	3, 14	bitri 278	. 2 ⊢ ([𝑀 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑀 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
16	1, 15	bitri 278	1 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑀 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  [wsbc 3742  ∪ ciun 4954  suc csuc 6379  ‘cfv 6555  ωcom 7898   predc-bnj14 35386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-v 3456  df-sbc 3743

This theorem is used by:  bnj600  35616  bnj908  35628  bnj964  35640  bnj999  35655