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Metamath Proof Explorer

Theorem tposfo2

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

Ref Expression
Assertion tposfo2 ( Rel 𝐴 → ( 𝐹 : 𝐴 –onto→ 𝐵 → tpos 𝐹 : ◡ 𝐴 –onto→ 𝐵 ) )

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 𝐴 ∧ 𝐹 Fn 𝐴 ) → Rel dom 𝐹 )
6 rntpos ⊢ ( Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹 )
7 5 6 syl ⊢ ( ( Rel 𝐴 ∧ 𝐹 Fn 𝐴 ) → ran tpos 𝐹 = ran 𝐹 )
8 7 eqeq1d ⊢ ( ( Rel 𝐴 ∧ 𝐹 Fn 𝐴 ) → ( ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵 ) )
9 8 biimprd ⊢ ( ( Rel 𝐴 ∧ 𝐹 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 : 𝐴 –onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) )
13 df-fo ⊢ ( tpos 𝐹 : ◡ 𝐴 –onto→ 𝐵 ↔ ( tpos 𝐹 Fn ◡ 𝐴 ∧ ran tpos 𝐹 = 𝐵 ) )
14 11 12 13 3imtr4g ⊢ ( Rel 𝐴 → ( 𝐹 : 𝐴 –onto→ 𝐵 → tpos 𝐹 : ◡ 𝐴 –onto→ 𝐵 ) )