Theorem f1eq3 5769

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

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 5706	. . 3 |- ( A = B -> ( F : C --> A <-> F : C --> B ) )
2	1	anbi1d 704	. 2 |- ( A = B -> ( ( F : C --> A /\ Fun `' F ) <-> ( F : C --> B /\ Fun `' F ) ) )
3		df-f1 5584	. 2 |- ( F : C -1-1-> A <-> ( F : C --> A /\ Fun `' F ) )
4		df-f1 5584	. 2 |- ( F : C -1-1-> B <-> ( F : C --> B /\ Fun `' F ) )
5	2, 3, 4	3bitr4g 288	1 |- ( A = B -> ( F : C -1-1-> A <-> F : C -1-1-> B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   `'ccnv 4991   Fun wfun 5573   -->wf 5575   -1-1->wf1 5576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-in 3476  df-ss 3483  df-f 5583  df-f1 5584

This theorem is referenced by:  f1oeq3  5800  f1eq123d  5802  tposf12  6970  brdomg  7516  pwfseq  9031  f1linds  18620  isuslgra  24006  isusgra  24007  isusgra0  24010  usgraop  24013  cusgraexilem2  24129  diaf1oN  35802