Theorem ovigg

Description: The value of an operation class abstraction. Compared with ovig , the condition ( x e. R /\ y e. S ) is removed. (Contributed by FL, 24-Mar-2007) (Revised by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	ovigg.1	$$⊢\left(x=A\wedge y=B\wedge z=C\right)\to \left(\phi \leftrightarrow \psi \right)$$
	ovigg.4	$$⊢{\exists }^{*}z\phi$$
	ovigg.5	$$⊢F=\left\{⟨⟨x,y⟩,z⟩|\phi \right\}$$
Assertion	ovigg	$$⊢\left(A\in V\wedge B\in W\wedge C\in X\right)\to \left(\psi \to AFB=C\right)$$

Proof

Step	Hyp	Ref	Expression
1		ovigg.1	$$⊢\left(x=A\wedge y=B\wedge z=C\right)\to \left(\phi \leftrightarrow \psi \right)$$
2		ovigg.4	$$⊢{\exists }^{*}z\phi$$
3		ovigg.5	$$⊢F=\left\{⟨⟨x,y⟩,z⟩|\phi \right\}$$
4	1	eloprabga	$$⊢\left(A\in V\wedge B\in W\wedge C\in X\right)\to \left(⟨⟨A,B⟩,C⟩\in \left\{⟨⟨x,y⟩,z⟩|\phi \right\}\leftrightarrow \psi \right)$$
5		df-ov	$$⊢AFB=F\left(⟨A,B⟩\right)$$
6	3	fveq1i	$$⊢F\left(⟨A,B⟩\right)=\left\{⟨⟨x,y⟩,z⟩|\phi \right\}\left(⟨A,B⟩\right)$$
7	5 6	eqtri	$$⊢AFB=\left\{⟨⟨x,y⟩,z⟩|\phi \right\}\left(⟨A,B⟩\right)$$
8	2	funoprab	$$⊢\mathrm{Fun}\left\{⟨⟨x,y⟩,z⟩|\phi \right\}$$
9		funopfv	$$⊢\mathrm{Fun}\left\{⟨⟨x,y⟩,z⟩|\phi \right\}\to \left(⟨⟨A,B⟩,C⟩\in \left\{⟨⟨x,y⟩,z⟩|\phi \right\}\to \left\{⟨⟨x,y⟩,z⟩|\phi \right\}\left(⟨A,B⟩\right)=C\right)$$
10	8 9	ax-mp	$$⊢⟨⟨A,B⟩,C⟩\in \left\{⟨⟨x,y⟩,z⟩|\phi \right\}\to \left\{⟨⟨x,y⟩,z⟩|\phi \right\}\left(⟨A,B⟩\right)=C$$
11	7 10	eqtrid	$$⊢⟨⟨A,B⟩,C⟩\in \left\{⟨⟨x,y⟩,z⟩|\phi \right\}\to AFB=C$$
12	4 11	biimtrrdi	$$⊢\left(A\in V\wedge B\in W\wedge C\in X\right)\to \left(\psi \to AFB=C\right)$$