Theorem bnj539 33891

Description: Technical lemma for bnj852 33921. 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 3837	. . 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 33740	. . . 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 3828	. . . . . . 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 3849	. . . . . . . 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 33738	. . . . . . 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 351	. . . . . 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 275	. . . . 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 3094	. . . 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 275	. . 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 275	. 2 ⊢ ([𝑀 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑀 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
16	1, 15	bitri 275	1 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑀 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  [wsbc 3777  ∪ ciun 4997  suc csuc 6364  ‘cfv 6541  ωcom 7852   predc-bnj14 33688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-v 3477  df-sbc 3778

This theorem is referenced by:  bnj600  33919  bnj908  33931  bnj964  33943  bnj999  33958