Theorem f1o4 3702

Description: Alternate definition of one-to-one onto function.

Assertion

Ref	Expression
f1o4	|- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))

Proof of Theorem f1o4

Step	Hyp	Ref	Expression
1		3anass 781	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
2		f1o2 3699	. 2 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
3		df-fn 3199	. . . 4 |- (`'F Fn B <-> (Fun `'F /\ dom `' F = B))
4		df-rn 3195	. . . . . 6 |- ran F = dom `' F
5	4	eqeq1i 1485	. . . . 5 |- (ran F = B <-> dom `' F = B)
6	5	anbi2i 482	. . . 4 |- ((Fun `'F /\ ran F = B) <-> (Fun `'F /\ dom `' F = B))
7	3, 6	bitr4 176	. . 3 |- (`'F Fn B <-> (Fun `'F /\ ran F = B))
8	7	anbi2i 482	. 2 |- ((F Fn A /\ `'F Fn B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
9	1, 2, 8	3bitr4 183	1 |- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958  `'ccnv 3175  dom cdm 3176  ran crn 3177  Fun wfun 3182   Fn wfn 3183  -1-1-onto->wf1o 3187

This theorem is referenced by:  f1oun 3712  f1o00 3720  f1osn 3725  f1ofveu 3888  curry1 4104  en2d 4406  sbthlem9 4461  cnvtr 10609

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-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-in 2054  df-ss 2056  df-rn 3195  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

Copyright terms: Public domain