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Metamath Proof Explorer

Theorem o1mptrcl

Description: Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016)

Ref Expression
Hypotheses o1add2.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
o1mptrcl.3 ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) )
Assertion o1mptrcl ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )

Proof

Step Hyp Ref Expression
1 o1add2.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
2 o1mptrcl.3 ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) )
3 o1f ⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℂ )
4 2 3 syl ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℂ )
5 1 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 )
6 dmmptg ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 )
7 5 6 syl ⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 )
8 7 feq2d ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℂ ↔ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) )
9 4 8 mpbid ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ )
10 9 fvmptelcdm ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )