Metamath Proof Explorer


Theorem tposfo2

Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015)

Ref Expression
Assertion tposfo2 ( Rel 𝐴 → ( 𝐹 : 𝐴onto𝐵 → tpos 𝐹 : 𝐴onto𝐵 ) )

Proof

Step Hyp Ref Expression
1 tposfn2 ( Rel 𝐴 → ( 𝐹 Fn 𝐴 → tpos 𝐹 Fn 𝐴 ) )
2 1 adantrd ( Rel 𝐴 → ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → tpos 𝐹 Fn 𝐴 ) )
3 fndm ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 )
4 3 releqd ( 𝐹 Fn 𝐴 → ( Rel dom 𝐹 ↔ Rel 𝐴 ) )
5 4 biimparc ( ( Rel 𝐴𝐹 Fn 𝐴 ) → Rel dom 𝐹 )
6 rntpos ( Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹 )
7 5 6 syl ( ( Rel 𝐴𝐹 Fn 𝐴 ) → ran tpos 𝐹 = ran 𝐹 )
8 7 eqeq1d ( ( Rel 𝐴𝐹 Fn 𝐴 ) → ( ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵 ) )
9 8 biimprd ( ( Rel 𝐴𝐹 Fn 𝐴 ) → ( ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵 ) )
10 9 expimpd ( Rel 𝐴 → ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → ran tpos 𝐹 = 𝐵 ) )
11 2 10 jcad ( Rel 𝐴 → ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → ( tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵 ) ) )
12 df-fo ( 𝐹 : 𝐴onto𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) )
13 df-fo ( tpos 𝐹 : 𝐴onto𝐵 ↔ ( tpos 𝐹 Fn 𝐴 ∧ ran tpos 𝐹 = 𝐵 ) )
14 11 12 13 3imtr4g ( Rel 𝐴 → ( 𝐹 : 𝐴onto𝐵 → tpos 𝐹 : 𝐴onto𝐵 ) )