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Metamath Proof Explorer

Theorem tposf1o2

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

Ref Expression
Assertion tposf1o2 ( Rel 𝐴 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → tpos 𝐹 : ◡ 𝐴 –1-1-onto→ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 tposf12 ⊢ ( Rel 𝐴 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → tpos 𝐹 : ◡ 𝐴 –1-1→ 𝐵 ) )
2 tposfo2 ⊢ ( Rel 𝐴 → ( 𝐹 : 𝐴 –onto→ 𝐵 → tpos 𝐹 : ◡ 𝐴 –onto→ 𝐵 ) )
3 1 2 anim12d ⊢ ( Rel 𝐴 → ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) → ( tpos 𝐹 : ◡ 𝐴 –1-1→ 𝐵 ∧ tpos 𝐹 : ◡ 𝐴 –onto→ 𝐵 ) ) )
4 df-f1o ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) )
5 df-f1o ⊢ ( tpos 𝐹 : ◡ 𝐴 –1-1-onto→ 𝐵 ↔ ( tpos 𝐹 : ◡ 𝐴 –1-1→ 𝐵 ∧ tpos 𝐹 : ◡ 𝐴 –onto→ 𝐵 ) )
6 3 4 5 3imtr4g ⊢ ( Rel 𝐴 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → tpos 𝐹 : ◡ 𝐴 –1-1-onto→ 𝐵 ) )