Theorem swapf1val

Description: The object part of the swap functor. See also swapf1vala . (Contributed by Zhi Wang, 7-Oct-2025)

	Ref	Expression
Hypotheses	swapfval.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑈 )
	swapfval.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑉 )
	swapf2fvala.s	⊢ 𝑆 = ( 𝐶 ×c 𝐷 )
	swapf2fvala.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	swapf1val.o	⊢ ( 𝜑 → ( 𝐶 swapF 𝐷 ) = ⟨ 𝑂 , 𝑃 ⟩ )
Assertion	swapf1val	⊢ ( 𝜑 → 𝑂 = ( 𝑥 ∈ 𝐵 ↦ ∪ ◡ { 𝑥 } ) )

Proof

Step	Hyp	Ref	Expression
1		swapfval.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑈 )
2		swapfval.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑉 )
3		swapf2fvala.s	⊢ 𝑆 = ( 𝐶 ×c 𝐷 )
4		swapf2fvala.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
5		swapf1val.o	⊢ ( 𝜑 → ( 𝐶 swapF 𝐷 ) = ⟨ 𝑂 , 𝑃 ⟩ )
6	5	fveq2d	⊢ ( 𝜑 → ( 1st ‘ ( 𝐶 swapF 𝐷 ) ) = ( 1st ‘ ⟨ 𝑂 , 𝑃 ⟩ ) )
7	1 2 3 4	swapf1vala	⊢ ( 𝜑 → ( 1st ‘ ( 𝐶 swapF 𝐷 ) ) = ( 𝑥 ∈ 𝐵 ↦ ∪ ◡ { 𝑥 } ) )
8	1 2	swapfelvv	⊢ ( 𝜑 → ( 𝐶 swapF 𝐷 ) ∈ ( V × V ) )
9	5 8	eqeltrrd	⊢ ( 𝜑 → ⟨ 𝑂 , 𝑃 ⟩ ∈ ( V × V ) )
10		opelxp	⊢ ( ⟨ 𝑂 , 𝑃 ⟩ ∈ ( V × V ) ↔ ( 𝑂 ∈ V ∧ 𝑃 ∈ V ) )
11	10	biimpi	⊢ ( ⟨ 𝑂 , 𝑃 ⟩ ∈ ( V × V ) → ( 𝑂 ∈ V ∧ 𝑃 ∈ V ) )
12		op1stg	⊢ ( ( 𝑂 ∈ V ∧ 𝑃 ∈ V ) → ( 1st ‘ ⟨ 𝑂 , 𝑃 ⟩ ) = 𝑂 )
13	9 11 12	3syl	⊢ ( 𝜑 → ( 1st ‘ ⟨ 𝑂 , 𝑃 ⟩ ) = 𝑂 )
14	6 7 13	3eqtr3rd	⊢ ( 𝜑 → 𝑂 = ( 𝑥 ∈ 𝐵 ↦ ∪ ◡ { 𝑥 } ) )