Description: Technical lemma for bnj150 . 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) (Revised by Mario Carneiro, 6-May-2015) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | bnj96.1 | ⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ } | |
| Assertion | bnj96 | ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → dom 𝐹 = 1o ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj96.1 | ⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ } | |
| 2 | bnj93 | ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ) | |
| 3 | dmsnopg | ⊢ ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V → dom { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ } = { ∅ } ) | |
| 4 | 2 3 | syl | ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → dom { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ } = { ∅ } ) |
| 5 | 1 | dmeqi | ⊢ dom 𝐹 = dom { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ } |
| 6 | df1o2 | ⊢ 1o = { ∅ } | |
| 7 | 4 5 6 | 3eqtr4g | ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → dom 𝐹 = 1o ) |