Theorem f1eq3 3668

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

Assertion

Ref	Expression
f1eq3	|- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))

Proof of Theorem f1eq3

Step	Hyp	Ref	Expression
1		feq3 3628	. . 3 |- (A = B -> (F:C-->A <-> F:C-->B))
2	1	anbi1d 619	. 2 |- (A = B -> ((F:C-->A /\ Fun `'F) <-> (F:C-->B /\ Fun `'F)))
3		df-f1 3201	. 2 |- (F:C-1-1->A <-> (F:C-->A /\ Fun `'F))
4		df-f1 3201	. 2 |- (F:C-1-1->B <-> (F:C-->B /\ Fun `'F))
5	2, 3, 4	3bitr4g 557	1 |- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958  `'ccnv 3175  Fun wfun 3182  -->wf 3184  -1-1->wf1 3185

This theorem is referenced by:  f1oeq3 3692  brdomg 4382

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056  df-f 3200  df-f1 3201

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