Theorem 1stval

Description: The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	1stval	⊢ ( 1st ‘ 𝐴 ) = ∪ dom { 𝐴 }

Proof

Step	Hyp	Ref	Expression
1		sneq	⊢ ( 𝑥 = 𝐴 → { 𝑥 } = { 𝐴 } )
2	1	dmeqd	⊢ ( 𝑥 = 𝐴 → dom { 𝑥 } = dom { 𝐴 } )
3	2	unieqd	⊢ ( 𝑥 = 𝐴 → ∪ dom { 𝑥 } = ∪ dom { 𝐴 } )
4		df-1st	⊢ 1st = ( 𝑥 ∈ V ↦ ∪ dom { 𝑥 } )
5		snex	⊢ { 𝐴 } ∈ V
6	5	dmex	⊢ dom { 𝐴 } ∈ V
7	6	uniex	⊢ ∪ dom { 𝐴 } ∈ V
8	3 4 7	fvmpt	⊢ ( 𝐴 ∈ V → ( 1st ‘ 𝐴 ) = ∪ dom { 𝐴 } )
9		fvprc	⊢ ( ¬ 𝐴 ∈ V → ( 1st ‘ 𝐴 ) = ∅ )
10		snprc	⊢ ( ¬ 𝐴 ∈ V ↔ { 𝐴 } = ∅ )
11	10	biimpi	⊢ ( ¬ 𝐴 ∈ V → { 𝐴 } = ∅ )
12	11	dmeqd	⊢ ( ¬ 𝐴 ∈ V → dom { 𝐴 } = dom ∅ )
13		dm0	⊢ dom ∅ = ∅
14	12 13	eqtrdi	⊢ ( ¬ 𝐴 ∈ V → dom { 𝐴 } = ∅ )
15	14	unieqd	⊢ ( ¬ 𝐴 ∈ V → ∪ dom { 𝐴 } = ∪ ∅ )
16		uni0	⊢ ∪ ∅ = ∅
17	15 16	eqtrdi	⊢ ( ¬ 𝐴 ∈ V → ∪ dom { 𝐴 } = ∅ )
18	9 17	eqtr4d	⊢ ( ¬ 𝐴 ∈ V → ( 1st ‘ 𝐴 ) = ∪ dom { 𝐴 } )
19	8 18	pm2.61i	⊢ ( 1st ‘ 𝐴 ) = ∪ dom { 𝐴 }