Theorem f1oeq3 3686

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

Assertion

Ref	Expression
f1oeq3	|- (A = B -> (F:C-1-1-onto->A <-> F:C-1-1-onto->B))

Proof of Theorem f1oeq3

Step	Hyp	Ref	Expression
1		f1eq3 3662	. . 3 |- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))
2		foeq3 3670	. . 3 |- (A = B -> (F:C-onto->A <-> F:C-onto->B))
3	1, 2	anbi12d 628	. 2 |- (A = B -> ((F:C-1-1->A /\ F:C-onto->A) <-> (F:C-1-1->B /\ F:C-onto->B)))
4		df-f1o 3197	. 2 |- (F:C-1-1-onto->A <-> (F:C-1-1->A /\ F:C-onto->A))
5		df-f1o 3197	. 2 |- (F:C-1-1-onto->B <-> (F:C-1-1->B /\ F:C-onto->B))
6	3, 4, 5	3bitr4g 555	1 |- (A = B -> (F:C-1-1-onto->A <-> F:C-1-1-onto->B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956  -1-1->wf1 3179  -onto->wfo 3180  -1-1-onto->wf1o 3181

This theorem is referenced by:  isoeq5 3891  ncanth 3908  breng 4375  idssen 4406  unfilem3 4550  icoshftf1olem 6410  reeff1o2 7427  nnenom 7498  unbenlem 7504  infxpidmlem2 7553  infxpidmlem3 7554  shftefif1olem 8741  adjbd1o 10018  elgiso 10398  symgval 10403  cayleylem3 10411  homeofval 10516

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-in 2051  df-ss 2053  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

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