Theorem f11o 3712

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 3711	. . 3 |- (F:A-->B <-> E.x(F:A-onto->x /\ x (_ B))
3	2	anbi1i 481	. 2 |- ((F:A-->B /\ Fun `'F) <-> (E.x(F:A-onto->x /\ x (_ B) /\ Fun `'F))
4		df-f1 3195	. 2 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
5		f1o3 3694	. . . . . 6 |- (F:A-1-1-onto->x <-> (F:A-onto->x /\ Fun `'F))
6	5	anbi1i 481	. . . . 5 |- ((F:A-1-1-onto->x /\ x (_ B) <-> ((F:A-onto->x /\ Fun `'F) /\ x (_ B))
7		an23 485	. . . . 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 1051	. . 3 |- (E.x(F:A-1-1-onto->x /\ x (_ B) <-> E.x((F:A-onto->x /\ x (_ B) /\ Fun `'F))
10		19.41v 1305	. . 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 958  E.wex 980  Vcvv 1811   (_ wss 2047  `'ccnv 3169  Fun wfun 3176  -->wf 3178  -1-1->wf1 3179  -onto->wfo 3180  -1-1-onto->wf1o 3181

This theorem is referenced by:  domen 4379

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-cnv 3186  df-dm 3188  df-rn 3189  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

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