Theorem f1eq2 5031

Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Assertion

Ref	Expression
f1eq2	|- (A = B -> ( F :A -1-1-> C <-> F :B -1-1-> C))

Proof of Theorem f1eq2

Step	Hyp	Ref	Expression
1		feq2 4974	. . 3 |- (A = B -> ( F :A --> C <-> F :B --> C))
2	1	anbi1d 438	. 2 |- (A = B -> (( F :A --> C /\ Fun `' F) <-> ( F :B --> C /\ Fun `' F)))
3		df-f1 4850	. 2 |- ( F :A -1-1-> C <-> ( F :A --> C /\ Fun `' F))
4		df-f1 4850	. 2 |- ( F :B -1-1-> C <-> ( F :B --> C /\ Fun `' F))
5	2, 3, 4	3bitr4g 212	1 |- (A = B -> ( F :A -1-1-> C <-> F :B -1-1-> C))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   `'ccnv 4287   Fun wfun 4839   -->wf 4841   -1-1->wf1 4842

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-fn 4848  df-f 4849  df-f1 4850

This theorem is referenced by:  f1oeq2  5061  f1eq123d  5064  brdomg  6165