Theorem f1oeq123d

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

	Ref	Expression
Hypotheses	f1eq123d.1	|- ( ph -> F = G )
	f1eq123d.2	|- ( ph -> A = B )
	f1eq123d.3	|- ( ph -> C = D )
Assertion	f1oeq123d	|- ( ph -> ( F : A -1-1-onto-> C <-> G : B -1-1-onto-> D ) )

Proof

Step	Hyp	Ref	Expression
1		f1eq123d.1	|- ( ph -> F = G )
2		f1eq123d.2	|- ( ph -> A = B )
3		f1eq123d.3	|- ( ph -> C = D )
4		f1oeq1	|- ( F = G -> ( F : A -1-1-onto-> C <-> G : A -1-1-onto-> C ) )
5	1 4	syl	|- ( ph -> ( 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	|- ( ph -> ( 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	|- ( ph -> ( G : B -1-1-onto-> C <-> G : B -1-1-onto-> D ) )
10	5 7 9	3bitrd	|- ( ph -> ( F : A -1-1-onto-> C <-> G : B -1-1-onto-> D ) )