Theorem ffoss 3696

Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145.

Hypothesis

Ref	Expression
f11o.1	|- F e. V

Assertion

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

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

Proof of Theorem ffoss

Step	Hyp	Ref	Expression
1		df-f 3184	. . . 4 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
2		fnforn 3662	. . . . 5 |- (F Fn A <-> F:A-onto->ran F)
3	2	anbi1i 480	. . . 4 |- ((F Fn A /\ ran F (_ B) <-> (F:A-onto->ran F /\ ran F (_ B))
4	1, 3	bitr 173	. . 3 |- (F:A-->B <-> (F:A-onto->ran F /\ ran F (_ B))
5		f11o.1	. . . . 5 |- F e. V
6		rnexg 3345	. . . . 5 |- (F e. V -> ran F e. V)
7	5, 6	ax-mp 7	. . . 4 |- ran F e. V
8		foeq3 3655	. . . . 5 |- (x = ran F -> (F:A-onto->x <-> F:A-onto->ran F))
9		sseq1 2072	. . . . 5 |- (x = ran F -> (x (_ B <-> ran F (_ B))
10	8, 9	anbi12d 626	. . . 4 |- (x = ran F -> ((F:A-onto->x /\ x (_ B) <-> (F:A-onto->ran F /\ ran F (_ B)))
11	7, 10	cla4ev 1860	. . 3 |- ((F:A-onto->ran F /\ ran F (_ B) -> E.x(F:A-onto->x /\ x (_ B))
12	4, 11	sylbi 199	. 2 |- (F:A-->B -> E.x(F:A-onto->x /\ x (_ B))
13		fss 3620	. . . 4 |- ((F:A-->x /\ x (_ B) -> F:A-->B)
14		fof 3657	. . . 4 |- (F:A-onto->x -> F:A-->x)
15	13, 14	sylan 448	. . 3 |- ((F:A-onto->x /\ x (_ B) -> F:A-->B)
16	15	19.23aiv 1290	. 2 |- (E.x(F:A-onto->x /\ x (_ B) -> F:A-->B)
17	12, 16	impbi 157	1 |- (F:A-->B <-> E.x(F:A-onto->x /\ x (_ B))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802   (_ wss 2037  ran crn 3161   Fn wfn 3167  -->wf 3168  -onto->wfo 3170

This theorem is referenced by:  f11o 3697

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-cnv 3176  df-dm 3178  df-rn 3179  df-f 3184  df-fo 3186

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