Theorem lo1mptrcl

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

	Ref	Expression
Hypotheses	o1add2.1	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
	lo1mptrcl.3	$$⊢\phi \to \left(x\in A⟼B\right)\in \le 𝑂\left(1\right)$$
Assertion	lo1mptrcl	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		o1add2.1	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
2		lo1mptrcl.3	$$⊢\phi \to \left(x\in A⟼B\right)\in \le 𝑂\left(1\right)$$
3		lo1f	$$⊢\left(x\in A⟼B\right)\in \le 𝑂\left(1\right)\to \left(x\in A⟼B\right):\mathrm{dom}\left(x\in A⟼B\right)⟶ℝ$$
4	2 3	syl	$$⊢\phi \to \left(x\in A⟼B\right):\mathrm{dom}\left(x\in A⟼B\right)⟶ℝ$$
5	1	ralrimiva	$$⊢\phi \to \forall x\in AB\in V$$
6		dmmptg	$$⊢\forall x\in AB\in V\to \mathrm{dom}\left(x\in A⟼B\right)=A$$
7	5 6	syl	$$⊢\phi \to \mathrm{dom}\left(x\in A⟼B\right)=A$$
8	7	feq2d	$$⊢\phi \to \left(\left(x\in A⟼B\right):\mathrm{dom}\left(x\in A⟼B\right)⟶ℝ\leftrightarrow \left(x\in A⟼B\right):A⟶ℝ\right)$$
9	4 8	mpbid	$$⊢\phi \to \left(x\in A⟼B\right):A⟶ℝ$$
10	9	fvmptelrn	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℝ$$