Theorem f1eq2 3658

Description: Equality theorem for one-to-one functions.

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 3618	. . 3 |- (A = B -> (F:A-->C <-> F:B-->C))
2	1	anbi1d 616	. 2 |- (A = B -> ((F:A-->C /\ Fun `'F) <-> (F:B-->C /\ Fun `'F)))
3		df-f1 3192	. 2 |- (F:A-1-1->C <-> (F:A-->C /\ Fun `'F))
4		df-f1 3192	. 2 |- (F:B-1-1->C <-> (F:B-->C /\ Fun `'F))
5	2, 3, 4	3bitr4g 554	1 |- (A = B -> (F:A-1-1->C <-> F:B-1-1->C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955  `'ccnv 3166  Fun wfun 3173  -->wf 3175  -1-1->wf1 3176

This theorem is referenced by:  f1oeq2 3682  brdomg 4371  unidom 4795

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-17 970  ax-4 972  ax-5o 974  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-cleq 1469  df-fn 3190  df-f 3191  df-f1 3192

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