Description: Technical lemma for bnj60 . 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 | bnj1307.1 | ⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } | |
| bnj1307.2 | ⊢ ( 𝑤 ∈ 𝐵 → ∀ 𝑥 𝑤 ∈ 𝐵 ) | ||
| Assertion | bnj1307 | ⊢ ( 𝑤 ∈ 𝐶 → ∀ 𝑥 𝑤 ∈ 𝐶 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1307.1 | ⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } | |
| 2 | bnj1307.2 | ⊢ ( 𝑤 ∈ 𝐵 → ∀ 𝑥 𝑤 ∈ 𝐵 ) | |
| 3 | 2 | nfcii | ⊢ Ⅎ 𝑥 𝐵 |
| 4 | nfv | ⊢ Ⅎ 𝑥 𝑓 Fn 𝑑 | |
| 5 | nfra1 | ⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) | |
| 6 | 4 5 | nfan | ⊢ Ⅎ 𝑥 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) |
| 7 | 3 6 | nfrex | ⊢ Ⅎ 𝑥 ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) |
| 8 | 7 | nfab | ⊢ Ⅎ 𝑥 { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
| 9 | 1 8 | nfcxfr | ⊢ Ⅎ 𝑥 𝐶 |
| 10 | 9 | nfcrii | ⊢ ( 𝑤 ∈ 𝐶 → ∀ 𝑥 𝑤 ∈ 𝐶 ) |