Theorem f1o4 2807

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 585	. 2 |- ((F Fn A /\ Fun `'F /\ ran F = B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
2		f1o2 2804	. 2 |- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
3		df-fn 2433	. . . 4 |- (`'F Fn B <-> (Fun `'F /\ dom `'F = B))
4		df-rn 2429	. . . . . 6 |- ran F = dom `'F
5	4	cleq1i 1108	. . . . 5 |- (ran F = B <-> dom `'F = B)
6	5	anbi2i 367	. . . 4 |- ((Fun `'F /\ ran F = B) <-> (Fun `'F /\ dom `'F = B))
7	3, 6	bitr4 154	. . 3 |- (`'F Fn B <-> (Fun `'F /\ ran F = B))
8	7	anbi2i 367	. 2 |- ((F Fn A /\ `'F Fn B) <-> (F Fn A /\ (Fun `'F /\ ran F = B)))
9	1, 2, 8	3bitr4 158	1 |- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))

Syntax hints:   <-> wb 127   /\ wa 196   /\ w3a 581   = wceq 1091  `'ccnv 2409  dom cdm 2410  ran crn 2411  Fun wfun 2416   Fn wfn 2417  -1-1-onto->wf1o 2421

This theorem is referenced by:  f1oun 2815  f1o00 2823  f1osn 2827  en2d 3303  sbthlem9 3357

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-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-rn 2429  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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