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Metamath Proof Explorer

Theorem ssimaexg

Description: The existence of a subimage. (Contributed by FL, 15-Apr-2007)

Ref Expression
Assertion ssimaexg ( ( 𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 imaeq2 ⊢ ( 𝑦 = 𝐴 → ( 𝐹 “ 𝑦 ) = ( 𝐹 “ 𝐴 ) )
2 1 sseq2d ⊢ ( 𝑦 = 𝐴 → ( 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ↔ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) )
3 2 anbi2d ⊢ ( 𝑦 = 𝐴 → ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ) ↔ ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) ) )
4 sseq2 ⊢ ( 𝑦 = 𝐴 → ( 𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴 ) )
5 4 anbi1d ⊢ ( 𝑦 = 𝐴 → ( ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ↔ ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) )
6 5 exbidv ⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ↔ ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) )
7 3 6 imbi12d ⊢ ( 𝑦 = 𝐴 → ( ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) ↔ ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) ) )
8 vex ⊢ 𝑦 ∈ V
9 8 ssimaex ⊢ ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) )
10 7 9 vtoclg ⊢ ( 𝐴 ∈ 𝐶 → ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) )
11 10 3impib ⊢ ( ( 𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) )