Theorem f1oeq123d

Description: Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017)

	Ref	Expression
Hypotheses	f1eq123d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
	f1eq123d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	f1eq123d.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	f1oeq123d	⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐺 : 𝐵 –1-1-onto→ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		f1eq123d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
2		f1eq123d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		f1eq123d.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
4		f1oeq1	⊢ ( 𝐹 = 𝐺 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐺 : 𝐴 –1-1-onto→ 𝐶 ) )
5	1 4	syl	⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐺 : 𝐴 –1-1-onto→ 𝐶 ) )
6		f1oeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐺 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐺 : 𝐵 –1-1-onto→ 𝐶 ) )
7	2 6	syl	⊢ ( 𝜑 → ( 𝐺 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐺 : 𝐵 –1-1-onto→ 𝐶 ) )
8		f1oeq3	⊢ ( 𝐶 = 𝐷 → ( 𝐺 : 𝐵 –1-1-onto→ 𝐶 ↔ 𝐺 : 𝐵 –1-1-onto→ 𝐷 ) )
9	3 8	syl	⊢ ( 𝜑 → ( 𝐺 : 𝐵 –1-1-onto→ 𝐶 ↔ 𝐺 : 𝐵 –1-1-onto→ 𝐷 ) )
10	5 7 9	3bitrd	⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐺 : 𝐵 –1-1-onto→ 𝐷 ) )