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 ) )