Metamath Proof Explorer


Theorem k0004lem2

Description: A mapping with a particular restricted range is also a mapping to that range. (Contributed by RP, 1-Apr-2021)

Ref Expression
Assertion k0004lem2 ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → ( ( 𝐹 ∈ ( 𝐵m 𝐴 ) ∧ ( 𝐹𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 ∈ ( 𝐶m 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 simp3 ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → 𝐶𝐵 )
2 sseqin2 ( 𝐶𝐵 ↔ ( 𝐵𝐶 ) = 𝐶 )
3 2 biimpi ( 𝐶𝐵 → ( 𝐵𝐶 ) = 𝐶 )
4 3 eqcomd ( 𝐶𝐵𝐶 = ( 𝐵𝐶 ) )
5 k0004lem1 ( 𝐶 = ( 𝐵𝐶 ) → ( ( 𝐹 : 𝐴𝐵 ∧ ( 𝐹𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 : 𝐴𝐶 ) )
6 1 4 5 3syl ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → ( ( 𝐹 : 𝐴𝐵 ∧ ( 𝐹𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 : 𝐴𝐶 ) )
7 simp2 ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → 𝐵𝑉 )
8 simp1 ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → 𝐴𝑈 )
9 7 8 elmapd ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → ( 𝐹 ∈ ( 𝐵m 𝐴 ) ↔ 𝐹 : 𝐴𝐵 ) )
10 9 anbi1d ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → ( ( 𝐹 ∈ ( 𝐵m 𝐴 ) ∧ ( 𝐹𝐴 ) ⊆ 𝐶 ) ↔ ( 𝐹 : 𝐴𝐵 ∧ ( 𝐹𝐴 ) ⊆ 𝐶 ) ) )
11 7 1 ssexd ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → 𝐶 ∈ V )
12 11 8 elmapd ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → ( 𝐹 ∈ ( 𝐶m 𝐴 ) ↔ 𝐹 : 𝐴𝐶 ) )
13 6 10 12 3bitr4d ( ( 𝐴𝑈𝐵𝑉𝐶𝐵 ) → ( ( 𝐹 ∈ ( 𝐵m 𝐴 ) ∧ ( 𝐹𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 ∈ ( 𝐶m 𝐴 ) ) )