Theorem bnj1039 35106

Description: Technical lemma for bnj69 35145. 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 3443	. . 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 3259	. . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
5	3, 4	nfxfr 1855	. . 3 ⊢ Ⅎ𝑖𝜓
6	2, 5	sbcgfi 3813	. 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 1542   ∈ wcel 2114  ∀wral 3050  [wsbc 3739  ∪ ciun 4945  suc csuc 6318  ‘cfv 6491  ωcom 7808   predc-bnj14 34823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2183  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-v 3441  df-sbc 3740

This theorem is referenced by:  bnj1128  35125