Metamath Proof Explorer


Theorem ffoss

Description: Relationship between a mapping and an onto mapping. Figure 38 of Enderton p. 145. (Contributed by NM, 10-May-1998)

Ref Expression
Hypothesis f11o.1 𝐹 ∈ V
Assertion ffoss ( 𝐹 : 𝐴𝐵 ↔ ∃ 𝑥 ( 𝐹 : 𝐴onto𝑥𝑥𝐵 ) )

Proof

Step Hyp Ref Expression
1 f11o.1 𝐹 ∈ V
2 df-f ( 𝐹 : 𝐴𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹𝐵 ) )
3 dffn4 ( 𝐹 Fn 𝐴𝐹 : 𝐴onto→ ran 𝐹 )
4 3 anbi1i ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹𝐵 ) ↔ ( 𝐹 : 𝐴onto→ ran 𝐹 ∧ ran 𝐹𝐵 ) )
5 2 4 bitri ( 𝐹 : 𝐴𝐵 ↔ ( 𝐹 : 𝐴onto→ ran 𝐹 ∧ ran 𝐹𝐵 ) )
6 1 rnex ran 𝐹 ∈ V
7 foeq3 ( 𝑥 = ran 𝐹 → ( 𝐹 : 𝐴onto𝑥𝐹 : 𝐴onto→ ran 𝐹 ) )
8 sseq1 ( 𝑥 = ran 𝐹 → ( 𝑥𝐵 ↔ ran 𝐹𝐵 ) )
9 7 8 anbi12d ( 𝑥 = ran 𝐹 → ( ( 𝐹 : 𝐴onto𝑥𝑥𝐵 ) ↔ ( 𝐹 : 𝐴onto→ ran 𝐹 ∧ ran 𝐹𝐵 ) ) )
10 6 9 spcev ( ( 𝐹 : 𝐴onto→ ran 𝐹 ∧ ran 𝐹𝐵 ) → ∃ 𝑥 ( 𝐹 : 𝐴onto𝑥𝑥𝐵 ) )
11 5 10 sylbi ( 𝐹 : 𝐴𝐵 → ∃ 𝑥 ( 𝐹 : 𝐴onto𝑥𝑥𝐵 ) )
12 fof ( 𝐹 : 𝐴onto𝑥𝐹 : 𝐴𝑥 )
13 fss ( ( 𝐹 : 𝐴𝑥𝑥𝐵 ) → 𝐹 : 𝐴𝐵 )
14 12 13 sylan ( ( 𝐹 : 𝐴onto𝑥𝑥𝐵 ) → 𝐹 : 𝐴𝐵 )
15 14 exlimiv ( ∃ 𝑥 ( 𝐹 : 𝐴onto𝑥𝑥𝐵 ) → 𝐹 : 𝐴𝐵 )
16 11 15 impbii ( 𝐹 : 𝐴𝐵 ↔ ∃ 𝑥 ( 𝐹 : 𝐴onto𝑥𝑥𝐵 ) )