Theorem f11o 3718

Description: Relationship between one-to-one and one-to-one onto function.

Hypothesis

Ref	Expression
f11o.1	|- F e. V

Assertion

Ref	Expression
f11o	|- (F:A-1-1->B <-> E.x(F:A-1-1-onto->x /\ x (_ B))

Distinct variable groups:   x,F   x,A   x,B

Proof of Theorem f11o

Step	Hyp	Ref	Expression
1		f11o.1	. . . 4 |- F e. V
2	1	ffoss 3717	. . 3 |- (F:A-->B <-> E.x(F:A-onto->x /\ x (_ B))
3	2	anbi1i 483	. 2 |- ((F:A-->B /\ Fun `'F) <-> (E.x(F:A-onto->x /\ x (_ B) /\ Fun `'F))
4		df-f1 3201	. 2 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
5		f1o3 3700	. . . . . 6 |- (F:A-1-1-onto->x <-> (F:A-onto->x /\ Fun `'F))
6	5	anbi1i 483	. . . . 5 |- ((F:A-1-1-onto->x /\ x (_ B) <-> ((F:A-onto->x /\ Fun `'F) /\ x (_ B))
7		an23 487	. . . . 5 |- (((F:A-onto->x /\ Fun `'F) /\ x (_ B) <-> ((F:A-onto->x /\ x (_ B) /\ Fun `'F))
8	6, 7	bitr 173	. . . 4 |- ((F:A-1-1-onto->x /\ x (_ B) <-> ((F:A-onto->x /\ x (_ B) /\ Fun `'F))
9	8	exbii 1053	. . 3 |- (E.x(F:A-1-1-onto->x /\ x (_ B) <-> E.x((F:A-onto->x /\ x (_ B) /\ Fun `'F))
10		19.41v 1307	. . 3 |- (E.x((F:A-onto->x /\ x (_ B) /\ Fun `'F) <-> (E.x(F:A-onto->x /\ x (_ B) /\ Fun `'F))
11	9, 10	bitr 173	. 2 |- (E.x(F:A-1-1-onto->x /\ x (_ B) <-> (E.x(F:A-onto->x /\ x (_ B) /\ Fun `'F))
12	3, 4, 11	3bitr4 183	1 |- (F:A-1-1->B <-> E.x(F:A-1-1-onto->x /\ x (_ B))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 960  E.wex 982  Vcvv 1814   (_ wss 2050  `'ccnv 3175  Fun wfun 3182  -->wf 3184  -1-1->wf1 3185  -onto->wfo 3186  -1-1-onto->wf1o 3187

This theorem is referenced by:  domen 4385

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-11 969  ax-12 970  ax-13 971  ax-14 972  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  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-cnv 3192  df-dm 3194  df-rn 3195  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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