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Metamath Proof Explorer

Theorem funimass3

Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv would be the special case of A being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006)

Ref Expression
Assertion funimass3 ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ( 𝐹 “ 𝐴 ) ⊆ 𝐵 ↔ 𝐴 ⊆ ( ◡ 𝐹 “ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 funimass4 ⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ( 𝐹 “ 𝐴 ) ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )
2 ssel ⊢ ( 𝐴 ⊆ dom 𝐹 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹 ) )
3 fvimacnv ⊢ ( ( Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ↔ 𝑥 ∈ ( ◡ 𝐹 “ 𝐵 ) ) )
4 3 ex ⊢ ( Fun 𝐹 → ( 𝑥 ∈ dom 𝐹 → ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ↔ 𝑥 ∈ ( ◡ 𝐹 “ 𝐵 ) ) ) )
5 2 4 syl9r ⊢ ( Fun 𝐹 → ( 𝐴 ⊆ dom 𝐹 → ( 𝑥 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ↔ 𝑥 ∈ ( ◡ 𝐹 “ 𝐵 ) ) ) ) )
6 5 imp31 ⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ↔ 𝑥 ∈ ( ◡ 𝐹 “ 𝐵 ) ) )
7 6 ralbidva ⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ( ◡ 𝐹 “ 𝐵 ) ) )
8 1 7 bitrd ⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ( 𝐹 “ 𝐴 ) ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ( ◡ 𝐹 “ 𝐵 ) ) )
9 dfss3 ⊢ ( 𝐴 ⊆ ( ◡ 𝐹 “ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ( ◡ 𝐹 “ 𝐵 ) )
10 8 9 bitr4di ⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ( 𝐹 “ 𝐴 ) ⊆ 𝐵 ↔ 𝐴 ⊆ ( ◡ 𝐹 “ 𝐵 ) ) )