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Metamath Proof Explorer

Theorem fo1st

Description: The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004) (Revised by Mario Carneiro, 8-Sep-2013)

Ref Expression
Assertion fo1st 1st : V –onto→ V

Proof

Step Hyp Ref Expression
1 vsnex ⊢ { 𝑥 } ∈ V
2 1 dmex ⊢ dom { 𝑥 } ∈ V
3 2 uniex ⊢ ∪ dom { 𝑥 } ∈ V
4 df-1st ⊢ 1st = ( 𝑥 ∈ V ↦ ∪ dom { 𝑥 } )
5 3 4 fnmpti ⊢ 1st Fn V
6 4 rnmpt ⊢ ran 1st = { 𝑦 ∣ ∃ 𝑥 ∈ V 𝑦 = ∪ dom { 𝑥 } }
7 vex ⊢ 𝑦 ∈ V
8 opex ⊢ ⟨ 𝑦 , 𝑦 ⟩ ∈ V
9 7 7 op1sta ⊢ ∪ dom { ⟨ 𝑦 , 𝑦 ⟩ } = 𝑦
10 9 eqcomi ⊢ 𝑦 = ∪ dom { ⟨ 𝑦 , 𝑦 ⟩ }
11 sneq ⊢ ( 𝑥 = ⟨ 𝑦 , 𝑦 ⟩ → { 𝑥 } = { ⟨ 𝑦 , 𝑦 ⟩ } )
12 11 dmeqd ⊢ ( 𝑥 = ⟨ 𝑦 , 𝑦 ⟩ → dom { 𝑥 } = dom { ⟨ 𝑦 , 𝑦 ⟩ } )
13 12 unieqd ⊢ ( 𝑥 = ⟨ 𝑦 , 𝑦 ⟩ → ∪ dom { 𝑥 } = ∪ dom { ⟨ 𝑦 , 𝑦 ⟩ } )
14 13 rspceeqv ⊢ ( ( ⟨ 𝑦 , 𝑦 ⟩ ∈ V ∧ 𝑦 = ∪ dom { ⟨ 𝑦 , 𝑦 ⟩ } ) → ∃ 𝑥 ∈ V 𝑦 = ∪ dom { 𝑥 } )
15 8 10 14 mp2an ⊢ ∃ 𝑥 ∈ V 𝑦 = ∪ dom { 𝑥 }
16 7 15 2th ⊢ ( 𝑦 ∈ V ↔ ∃ 𝑥 ∈ V 𝑦 = ∪ dom { 𝑥 } )
17 16 eqabi ⊢ V = { 𝑦 ∣ ∃ 𝑥 ∈ V 𝑦 = ∪ dom { 𝑥 } }
18 6 17 eqtr4i ⊢ ran 1st = V
19 df-fo ⊢ ( 1st : V –onto→ V ↔ ( 1st Fn V ∧ ran 1st = V ) )
20 5 18 19 mpbir2an ⊢ 1st : V –onto→ V