Metamath Proof Explorer


Theorem ssimaexg

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

Ref Expression
Assertion ssimaexg ( ( 𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ ( 𝐹𝐴 ) ) → ∃ 𝑥 ( 𝑥𝐴𝐵 = ( 𝐹𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 imaeq2 ( 𝑦 = 𝐴 → ( 𝐹𝑦 ) = ( 𝐹𝐴 ) )
2 1 sseq2d ( 𝑦 = 𝐴 → ( 𝐵 ⊆ ( 𝐹𝑦 ) ↔ 𝐵 ⊆ ( 𝐹𝐴 ) ) )
3 2 anbi2d ( 𝑦 = 𝐴 → ( ( Fun 𝐹𝐵 ⊆ ( 𝐹𝑦 ) ) ↔ ( Fun 𝐹𝐵 ⊆ ( 𝐹𝐴 ) ) ) )
4 sseq2 ( 𝑦 = 𝐴 → ( 𝑥𝑦𝑥𝐴 ) )
5 4 anbi1d ( 𝑦 = 𝐴 → ( ( 𝑥𝑦𝐵 = ( 𝐹𝑥 ) ) ↔ ( 𝑥𝐴𝐵 = ( 𝐹𝑥 ) ) ) )
6 5 exbidv ( 𝑦 = 𝐴 → ( ∃ 𝑥 ( 𝑥𝑦𝐵 = ( 𝐹𝑥 ) ) ↔ ∃ 𝑥 ( 𝑥𝐴𝐵 = ( 𝐹𝑥 ) ) ) )
7 3 6 imbi12d ( 𝑦 = 𝐴 → ( ( ( Fun 𝐹𝐵 ⊆ ( 𝐹𝑦 ) ) → ∃ 𝑥 ( 𝑥𝑦𝐵 = ( 𝐹𝑥 ) ) ) ↔ ( ( Fun 𝐹𝐵 ⊆ ( 𝐹𝐴 ) ) → ∃ 𝑥 ( 𝑥𝐴𝐵 = ( 𝐹𝑥 ) ) ) ) )
8 vex 𝑦 ∈ V
9 8 ssimaex ( ( Fun 𝐹𝐵 ⊆ ( 𝐹𝑦 ) ) → ∃ 𝑥 ( 𝑥𝑦𝐵 = ( 𝐹𝑥 ) ) )
10 7 9 vtoclg ( 𝐴𝐶 → ( ( Fun 𝐹𝐵 ⊆ ( 𝐹𝐴 ) ) → ∃ 𝑥 ( 𝑥𝐴𝐵 = ( 𝐹𝑥 ) ) ) )
11 10 3impib ( ( 𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ ( 𝐹𝐴 ) ) → ∃ 𝑥 ( 𝑥𝐴𝐵 = ( 𝐹𝑥 ) ) )