Theorem foeq2 6112

Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
foeq2	|- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )

Proof of Theorem foeq2

Step	Hyp	Ref	Expression
1		fneq2 5980	. . 3 |- ( A = B -> ( F Fn A <-> F Fn B ) )
2	1	anbi1d 741	. 2 |- ( A = B -> ( ( F Fn A /\ ran F = C ) <-> ( F Fn B /\ ran F = C ) ) )
3		df-fo 5894	. 2 |- ( F : A -onto-> C <-> ( F Fn A /\ ran F = C ) )
4		df-fo 5894	. 2 |- ( F : B -onto-> C <-> ( F Fn B /\ ran F = C ) )
5	2, 3, 4	3bitr4g 303	1 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   ran crn 5115   Fn wfn 5883   -onto->wfo 5886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-fn 5891  df-fo 5894

This theorem is referenced by:  f1oeq2  6128  foeq123d  6132  tposfo  7379  brwdom  8472  brwdom2  8478  canthwdom  8484  cfslb2n  9090  fodomg  9345  0ramcl  15727  ghmcyg  18297  txcmpb  21447  qtoptopon  21507  opidon2OLD  33653