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 ) )