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Metamath Proof Explorer

Theorem ovelrn

Description: A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007) (Revised by Mario Carneiro, 30-Jan-2014)

Ref Expression
Assertion ovelrn ( 𝐹 Fn ( 𝐴 × 𝐵 ) → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 𝐹 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 fnrnov ⊢ ( 𝐹 Fn ( 𝐴 × 𝐵 ) → ran 𝐹 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝑥 𝐹 𝑦 ) } )
2 1 eleq2d ⊢ ( 𝐹 Fn ( 𝐴 × 𝐵 ) → ( 𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝑥 𝐹 𝑦 ) } ) )
3 ovex ⊢ ( 𝑥 𝐹 𝑦 ) ∈ V
4 eleq1 ⊢ ( 𝐶 = ( 𝑥 𝐹 𝑦 ) → ( 𝐶 ∈ V ↔ ( 𝑥 𝐹 𝑦 ) ∈ V ) )
5 3 4 mpbiri ⊢ ( 𝐶 = ( 𝑥 𝐹 𝑦 ) → 𝐶 ∈ V )
6 5 rexlimivw ⊢ ( ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 𝐹 𝑦 ) → 𝐶 ∈ V )
7 6 rexlimivw ⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 𝐹 𝑦 ) → 𝐶 ∈ V )
8 eqeq1 ⊢ ( 𝑧 = 𝐶 → ( 𝑧 = ( 𝑥 𝐹 𝑦 ) ↔ 𝐶 = ( 𝑥 𝐹 𝑦 ) ) )
9 8 2rexbidv ⊢ ( 𝑧 = 𝐶 → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝑥 𝐹 𝑦 ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 𝐹 𝑦 ) ) )
10 7 9 elab3 ⊢ ( 𝐶 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑧 = ( 𝑥 𝐹 𝑦 ) } ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 𝐹 𝑦 ) )
11 2 10 bitrdi ⊢ ( 𝐹 Fn ( 𝐴 × 𝐵 ) → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 𝐹 𝑦 ) ) )