Theorem fconstfv 2903

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 2902.

Assertion

Ref	Expression
fconstfv	|- (F:A-->{B} <-> (F Fn A /\ A.x e. A (F` x) = B))

Distinct variable group(s):   x,A   x,B   x,F

Proof of Theorem fconstfv

Step	Hyp	Ref	Expression
1		ffn 2752	. . 3 |- (F:A-->{B} -> F Fn A)
2		fvconst 2899	. . . . 5 |- ((F:A-->{B} /\ x e. A) -> (F` x) = B)
3	2	exp 291	. . . 4 |- (F:A-->{B} -> (x e. A -> (F` x) = B))
4	3	r19.21aiv 1259	. . 3 |- (F:A-->{B} -> A.x e. A (F` x) = B)
5	1, 4	jca 236	. 2 |- (F:A-->{B} -> (F Fn A /\ A.x e. A (F` x) = B))
6		fneq2 2719	. . . . . . 7 |- (A = (/) -> (F Fn A <-> F Fn (/)))
7		fn0 2739	. . . . . . 7 |- (F Fn (/) <-> F = (/))
8	6, 7	syl6bb 414	. . . . . 6 |- (A = (/) -> (F Fn A <-> F = (/)))
9		f0 2772	. . . . . . 7 |- (/):(/)-->{B}
10		feq1 2748	. . . . . . 7 |- (F = (/) -> (F:(/)-->{B} <-> (/):(/)-->{B}))
11	9, 10	mpbiri 169	. . . . . 6 |- (F = (/) -> F:(/)-->{B})
12	8, 11	syl6bi 187	. . . . 5 |- (A = (/) -> (F Fn A -> F:(/)-->{B}))
13		feq2 2749	. . . . 5 |- (A = (/) -> (F:A-->{B} <-> F:(/)-->{B}))
14	12, 13	sylibrd 179	. . . 4 |- (A = (/) -> (F Fn A -> F:A-->{B}))
15	14	adantrd 308	. . 3 |- (A = (/) -> ((F Fn A /\ A.x e. A (F` x) = B) -> F:A-->{B}))
16		fvelrn 2883	. . . . . . . . . 10 |- (F Fn A -> (y e. ran F <-> E.z e. A (F` z) = y))
17		fveq2 2832	. . . . . . . . . . . . . . . 16 |- (x = z -> (F` x) = (F` z))
18	17	cleq1d 1109	. . . . . . . . . . . . . . 15 |- (x = z -> ((F` x) = B <-> (F` z) = B))
19	18	rcla4v 1402	. . . . . . . . . . . . . 14 |- (A.x e. A (F` x) = B -> (z e. A -> (F` z) = B))
20	19	imp 277	. . . . . . . . . . . . 13 |- ((A.x e. A (F` x) = B /\ z e. A) -> (F` z) = B)
21	20	cleq1d 1109	. . . . . . . . . . . 12 |- ((A.x e. A (F` x) = B /\ z e. A) -> ((F` z) = y <-> B = y))
22	21	birexdva 1216	. . . . . . . . . . 11 |- (A.x e. A (F` x) = B -> (E.z e. A (F` z) = y <-> E.z e. A B = y))
23		r19.9rzv 1768	. . . . . . . . . . . 12 |- (-. A = (/) -> (B = y <-> E.z e. A B = y))
24	23	bicomd 399	. . . . . . . . . . 11 |- (-. A = (/) -> (E.z e. A B = y <-> B = y))
25	22, 24	sylan9bbr 419	. . . . . . . . . 10 |- ((-. A = (/) /\ A.x e. A (F` x) = B) -> (E.z e. A (F` z) = y <-> B = y))
26	16, 25	sylan9bbr 419	. . . . . . . . 9 |- (((-. A = (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> (y e. ran F <-> B = y))
27		elsn 1820	. . . . . . . . . 10 |- (y e. {B} <-> y = B)
28		cleqcom 1103	. . . . . . . . . 10 |- (y = B <-> B = y)
29	27, 28	bitr2 152	. . . . . . . . 9 |- (B = y <-> y e. {B})
30	26, 29	syl6bb 414	. . . . . . . 8 |- (((-. A = (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> (y e. ran F <-> y e. {B}))
31	30	cleqrd 1100	. . . . . . 7 |- (((-. A = (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> ran F = {B})
32	31	an1rs 373	. . . . . 6 |- (((-. A = (/) /\ F Fn A) /\ A.x e. A (F` x) = B) -> ran F = {B})
33	32	exp31 293	. . . . 5 |- (-. A = (/) -> (F Fn A -> (A.x e. A (F` x) = B -> ran F = {B})))
34	33	imdistand 342	. . . 4 |- (-. A = (/) -> ((F Fn A /\ A.x e. A (F` x) = B) -> (F Fn A /\ ran F = {B})))
35		df-fo 2436	. . . . 5 |- (F:A-onto->{B} <-> (F Fn A /\ ran F = {B}))
36		fof 2788	. . . . 5 |- (F:A-onto->{B} -> F:A-->{B})
37	35, 36	sylbir 176	. . . 4 |- ((F Fn A /\ ran F = {B}) -> F:A-->{B})
38	34, 37	syl6 23	. . 3 |- (-. A = (/) -> ((F Fn A /\ A.x e. A (F` x) = B) -> F:A-->{B}))
39	15, 38	pm2.61i 110	. 2 |- ((F Fn A /\ A.x e. A (F` x) = B) -> F:A-->{B})
40	5, 39	impbi 139	1 |- (F:A-->{B} <-> (F Fn A /\ A.x e. A (F` x) = B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  (/)c0 1707  {csn 1808  ran crn 2411   Fn wfn 2417  -->wf 2418  -onto->wfo 2420  ` cfv 2422

This theorem is referenced by:  ho0 5612

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fo 2436  df-fv 2438

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