Theorem bnj1039 35304

Description: Technical lemma for bnj69 35343. 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
bnj1039.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1039.2	⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓)

Assertion

Ref	Expression
bnj1039	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Proof of Theorem bnj1039

Step	Hyp	Ref	Expression
1		bnj1039.2	. 2 ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓)
2		vex 3465	. . 3 ⊢ 𝑗 ∈ V
3		bnj1039.1	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4		nfra1 3295	. . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
5	3, 4	nfxfr 1880	. . 3 ⊢ Ⅎ𝑖𝜓
6	2, 5	sbcgfi 3824	. 2 ⊢ ([𝑗 / 𝑖]𝜓 ↔ 𝜓)
7	1, 6, 3	3bitri 300	1 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∀wral 3085  [wsbc 3751  ∪ ciun 4958  suc csuc 6363  ‘cfv 6537  ωcom 7862   predc-bnj14 35022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-v 3463  df-sbc 3752

This theorem is referenced by:  bnj1128  35323