Theorem dffo4 3811

Description: Alternate definition of an onto mapping.

Assertion

Ref	Expression
dffo4	|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A xFy))

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

Proof of Theorem dffo4

Step	Hyp	Ref	Expression
1		dffo2 3666	. . 3 |- (F:A-onto->B <-> (F:A-->B /\ ran F = B))
2		pm3.26 319	. . . 4 |- ((F:A-->B /\ ran F = B) -> F:A-->B)
3		eleq2 1532	. . . . . . . . 9 |- (ran F = B -> (y e. ran F <-> y e. B))
4		visset 1809	. . . . . . . . . 10 |- y e. V
5	4	elrn 3344	. . . . . . . . 9 |- (y e. ran F <-> E.x xFy)
6	3, 5	syl5bbr 533	. . . . . . . 8 |- (ran F = B -> (E.x xFy <-> y e. B))
7	6	biimpar 417	. . . . . . 7 |- ((ran F = B /\ y e. B) -> E.x xFy)
8	7	adantll 392	. . . . . 6 |- (((F:A-->B /\ ran F = B) /\ y e. B) -> E.x xFy)
9		ffn 3619	. . . . . . . . . . 11 |- (F:A-->B -> F Fn A)
10		fnbr 3582	. . . . . . . . . . . 12 |- ((F Fn A /\ xFy) -> x e. A)
11	10	ex 373	. . . . . . . . . . 11 |- (F Fn A -> (xFy -> x e. A))
12	9, 11	syl 10	. . . . . . . . . 10 |- (F:A-->B -> (xFy -> x e. A))
13	12	ancrd 299	. . . . . . . . 9 |- (F:A-->B -> (xFy -> (x e. A /\ xFy)))
14	13	19.22dv 1288	. . . . . . . 8 |- (F:A-->B -> (E.x xFy -> E.x(x e. A /\ xFy)))
15		df-rex 1647	. . . . . . . 8 |- (E.x e. A xFy <-> E.x(x e. A /\ xFy))
16	14, 15	syl6ibr 213	. . . . . . 7 |- (F:A-->B -> (E.x xFy -> E.x e. A xFy))
17	16	ad2antrr 404	. . . . . 6 |- (((F:A-->B /\ ran F = B) /\ y e. B) -> (E.x xFy -> E.x e. A xFy))
18	8, 17	mpd 26	. . . . 5 |- (((F:A-->B /\ ran F = B) /\ y e. B) -> E.x e. A xFy)
19	18	r19.21aiva 1711	. . . 4 |- ((F:A-->B /\ ran F = B) -> A.y e. B E.x e. A xFy)
20	2, 19	jca 288	. . 3 |- ((F:A-->B /\ ran F = B) -> (F:A-->B /\ A.y e. B E.x e. A xFy))
21	1, 20	sylbi 199	. 2 |- (F:A-onto->B -> (F:A-->B /\ A.y e. B E.x e. A xFy))
22	4	fnbrfvb 3744	. . . . . . . . 9 |- ((F Fn A /\ x e. A) -> ((F` x) = y <-> xFy))
23	22	biimprd 154	. . . . . . . 8 |- ((F Fn A /\ x e. A) -> (xFy -> (F` x) = y))
24		eqcom 1474	. . . . . . . 8 |- ((F` x) = y <-> y = (F` x))
25	23, 24	syl6ib 212	. . . . . . 7 |- ((F Fn A /\ x e. A) -> (xFy -> y = (F` x)))
26	25, 9	sylan 448	. . . . . 6 |- ((F:A-->B /\ x e. A) -> (xFy -> y = (F` x)))
27	26	r19.22dva 1736	. . . . 5 |- (F:A-->B -> (E.x e. A xFy -> E.x e. A y = (F` x)))
28	27	r19.20sdv 1707	. . . 4 |- (F:A-->B -> (A.y e. B E.x e. A xFy -> A.y e. B E.x e. A y = (F` x)))
29	28	imdistani 443	. . 3 |- ((F:A-->B /\ A.y e. B E.x e. A xFy) -> (F:A-->B /\ A.y e. B E.x e. A y = (F` x)))
30		dffo3 3810	. . 3 |- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A y = (F` x)))
31	29, 30	sylibr 200	. 2 |- ((F:A-->B /\ A.y e. B E.x e. A xFy) -> F:A-onto->B)
32	21, 31	impbi 157	1 |- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A xFy))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  A.wral 1642  E.wrex 1643   class class class wbr 2614  ran crn 3166   Fn wfn 3172  -->wf 3173  -onto->wfo 3175  ` cfv 3177

This theorem is referenced by:  dffo5 3812  exfo 3813  brdom3 4781

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193

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