Theorem ffoss 3711

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 3194	. . . 4 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
2		fnforn 3677	. . . . 5 |- (F Fn A <-> F:A-onto->ran F)
3	2	anbi1i 481	. . . 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 3361	. . . 4 |- ran F e. V
7		foeq3 3670	. . . . 5 |- (x = ran F -> (F:A-onto->x <-> F:A-onto->ran F))
8		sseq1 2082	. . . . 5 |- (x = ran F -> (x (_ B <-> ran F (_ B))
9	7, 8	anbi12d 628	. . . 4 |- (x = ran F -> ((F:A-onto->x /\ x (_ B) <-> (F:A-onto->ran F /\ ran F (_ B)))
10	6, 9	cla4ev 1869	. . 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 3635	. . . 4 |- ((F:A-->x /\ x (_ B) -> F:A-->B)
13		fof 3672	. . . 4 |- (F:A-onto->x -> F:A-->x)
14	12, 13	sylan 448	. . 3 |- ((F:A-onto->x /\ x (_ B) -> F:A-->B)
15	14	19.23aiv 1295	. 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 956   e. wcel 958  E.wex 980  Vcvv 1811   (_ wss 2047  ran crn 3171   Fn wfn 3177  -->wf 3178  -onto->wfo 3180

This theorem is referenced by:  f11o 3712

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-fo 3196

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