Description: Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013) (Revised by Mario Carneiro, 20-Aug-2014)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | rexrn.1 | ⊢ ( 𝑥 = ( 𝐹 ‘ 𝑦 ) → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | ralrn | ⊢ ( 𝐹 Fn 𝐴 → ( ∀ 𝑥 ∈ ran 𝐹 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rexrn.1 | ⊢ ( 𝑥 = ( 𝐹 ‘ 𝑦 ) → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | fvexd | ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) ∈ V ) | |
| 3 | fvelrnb | ⊢ ( 𝐹 Fn 𝐴 → ( 𝑥 ∈ ran 𝐹 ↔ ∃ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = 𝑥 ) ) | |
| 4 | eqcom | ⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑥 ↔ 𝑥 = ( 𝐹 ‘ 𝑦 ) ) | |
| 5 | 4 | rexbii | ⊢ ( ∃ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝐹 ‘ 𝑦 ) ) |
| 6 | 3 5 | bitrdi | ⊢ ( 𝐹 Fn 𝐴 → ( 𝑥 ∈ ran 𝐹 ↔ ∃ 𝑦 ∈ 𝐴 𝑥 = ( 𝐹 ‘ 𝑦 ) ) ) |
| 7 | 1 | adantl | ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑥 = ( 𝐹 ‘ 𝑦 ) ) → ( 𝜑 ↔ 𝜓 ) ) |
| 8 | 2 6 7 | ralxfr2d | ⊢ ( 𝐹 Fn 𝐴 → ( ∀ 𝑥 ∈ ran 𝐹 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 𝜓 ) ) |