Theorem f1oeq3 2797

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 2778	. . 3 |- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))
2		foeq3 2786	. . 3 |- (A = B -> (F:C-onto->A <-> F:C-onto->B))
3	1, 2	anbi12d 476	. 2 |- (A = B -> ((F:C-1-1->A /\ F:C-onto->A) <-> (F:C-1-1->B /\ F:C-onto->B)))
4		df-f1o 2437	. 2 |- (F:C-1-1-onto->A <-> (F:C-1-1->A /\ F:C-onto->A))
5		df-f1o 2437	. 2 |- (F:C-1-1-onto->B <-> (F:C-1-1->B /\ F:C-onto->B))
6	3, 4, 5	3bitr4g 428	1 |- (A = B -> (F:C-1-1-onto->A <-> F:C-1-1-onto->B))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091  -1-1->wf1 2419  -onto->wfo 2420  -1-1-onto->wf1o 2421

This theorem is referenced by:  isoeq5 2929  ncanth 2946  breng 3280  idssen 3309  unfilem3 3440  nnenom 4926  infxpidmlem2 4934  infxpidmlem3 4935

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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