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Metamath Proof Explorer

Theorem f1mptrn

Description: Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020)

Ref Expression
Hypotheses f1mptrn.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
f1mptrn.2 ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → ∃! 𝑥 ∈ 𝐴 𝑦 = 𝐵 )
Assertion f1mptrn ( 𝜑 → Fun ◡ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 f1mptrn.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
2 f1mptrn.2 ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → ∃! 𝑥 ∈ 𝐴 𝑦 = 𝐵 )
3 1 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 )
4 2 ralrimiva ⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐶 ∃! 𝑥 ∈ 𝐴 𝑦 = 𝐵 )
5 eqid ⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
6 5 f1ompt ⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 –1-1-onto→ 𝐶 ↔ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ∃! 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) )
7 dff1o2 ⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 –1-1-onto→ 𝐶 ↔ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 ∧ Fun ◡ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐶 ) )
8 7 simp2bi ⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 –1-1-onto→ 𝐶 → Fun ◡ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
9 6 8 sylbir ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀ 𝑦 ∈ 𝐶 ∃! 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) → Fun ◡ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
10 3 4 9 syl2anc ⊢ ( 𝜑 → Fun ◡ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )