Theorem bnj1039 34983

Description: Technical lemma for bnj69 35022. 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 3440	. . 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 3256	. . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
5	3, 4	nfxfr 1854	. . 3 ⊢ Ⅎ𝑖𝜓
6	2, 5	sbcgfi 3810	. 2 ⊢ ([𝑗 / 𝑖]𝜓 ↔ 𝜓)
7	1, 6, 3	3bitri 297	1 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∀wral 3047  [wsbc 3736  ∪ ciun 4939  suc csuc 6308  ‘cfv 6481  ωcom 7796   predc-bnj14 34700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-sbc 3737

This theorem is referenced by:  bnj1128  35002