Description: Technical lemma for bnj69 . 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.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | 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 | bnj1039 | ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1039.1 | ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) | |
| 2 | bnj1039.2 | ⊢ ( 𝜓′ ↔ [ 𝑗 / 𝑖 ] 𝜓 ) | |
| 3 | vex | ⊢ 𝑗 ∈ V | |
| 4 | nfra1 | ⊢ Ⅎ 𝑖 ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) | |
| 5 | 1 4 | nfxfr | ⊢ Ⅎ 𝑖 𝜓 |
| 6 | 3 5 | sbcgfi | ⊢ ( [ 𝑗 / 𝑖 ] 𝜓 ↔ 𝜓 ) |
| 7 | 2 6 1 | 3bitri | ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |