Theorem ffoss 3717

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 3200	. . . 4 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
2		fnforn 3683	. . . . 5 |- (F Fn A <-> F:A-onto->ran F)
3	2	anbi1i 483	. . . 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	5	rnex 3367	. . . 4 |- ran F e. V
7		foeq3 3676	. . . . 5 |- (x = ran F -> (F:A-onto->x <-> F:A-onto->ran F))
8		sseq1 2085	. . . . 5 |- (x = ran F -> (x (_ B <-> ran F (_ B))
9	7, 8	anbi12d 630	. . . 4 |- (x = ran F -> ((F:A-onto->x /\ x (_ B) <-> (F:A-onto->ran F /\ ran F (_ B)))
10	6, 9	cla4ev 1872	. . 3 |- ((F:A-onto->ran F /\ ran F (_ B) -> E.x(F:A-onto->x /\ x (_ B))
11	4, 10	sylbi 199	. 2 |- (F:A-->B -> E.x(F:A-onto->x /\ x (_ B))
12		fss 3641	. . . 4 |- ((F:A-->x /\ x (_ B) -> F:A-->B)
13		fof 3678	. . . 4 |- (F:A-onto->x -> F:A-->x)
14	12, 13	sylan 450	. . 3 |- ((F:A-onto->x /\ x (_ B) -> F:A-->B)
15	14	19.23aiv 1297	. 2 |- (E.x(F:A-onto->x /\ x (_ B) -> F:A-->B)
16	11, 15	impbi 157	1 |- (F:A-->B <-> E.x(F:A-onto->x /\ x (_ B))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  Vcvv 1814   (_ wss 2050  ran crn 3177   Fn wfn 3183  -->wf 3184  -onto->wfo 3186

This theorem is referenced by:  f11o 3718

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-fo 3202

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