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Metamath Proof Explorer

Theorem 2ndval

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

Ref Expression
Assertion 2ndval ( 2nd ‘ 𝐴 ) = ∪ ran { 𝐴 }

Proof

Step Hyp Ref Expression
1 sneq ⊢ ( 𝑥 = 𝐴 → { 𝑥 } = { 𝐴 } )
2 1 rneqd ⊢ ( 𝑥 = 𝐴 → ran { 𝑥 } = ran { 𝐴 } )
3 2 unieqd ⊢ ( 𝑥 = 𝐴 → ∪ ran { 𝑥 } = ∪ ran { 𝐴 } )
4 df-2nd ⊢ 2nd = ( 𝑥 ∈ V ↦ ∪ ran { 𝑥 } )
5 snex ⊢ { 𝐴 } ∈ V
6 5 rnex ⊢ ran { 𝐴 } ∈ V
7 6 uniex ⊢ ∪ ran { 𝐴 } ∈ V
8 3 4 7 fvmpt ⊢ ( 𝐴 ∈ V → ( 2nd ‘ 𝐴 ) = ∪ ran { 𝐴 } )
9 fvprc ⊢ ( ¬ 𝐴 ∈ V → ( 2nd ‘ 𝐴 ) = ∅ )
10 snprc ⊢ ( ¬ 𝐴 ∈ V ↔ { 𝐴 } = ∅ )
11 10 biimpi ⊢ ( ¬ 𝐴 ∈ V → { 𝐴 } = ∅ )
12 11 rneqd ⊢ ( ¬ 𝐴 ∈ V → ran { 𝐴 } = ran ∅ )
13 rn0 ⊢ ran ∅ = ∅
14 12 13 eqtrdi ⊢ ( ¬ 𝐴 ∈ V → ran { 𝐴 } = ∅ )
15 14 unieqd ⊢ ( ¬ 𝐴 ∈ V → ∪ ran { 𝐴 } = ∪ ∅ )
16 uni0 ⊢ ∪ ∅ = ∅
17 15 16 eqtrdi ⊢ ( ¬ 𝐴 ∈ V → ∪ ran { 𝐴 } = ∅ )
18 9 17 eqtr4d ⊢ ( ¬ 𝐴 ∈ V → ( 2nd ‘ 𝐴 ) = ∪ ran { 𝐴 } )
19 8 18 pm2.61i ⊢ ( 2nd ‘ 𝐴 ) = ∪ ran { 𝐴 }