Theorem fconstfv 5457

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5455. (Contributed by NM, 27-Aug-2004.)

Assertion

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

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

Proof of Theorem fconstfv

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5224	. . 3 |- (F:A-->{B} -> F Fn A)
2		fvconst 5441	. . . 4 |- ((F:A-->{B} /\ x e. A) -> (F` x) = B)
3	2	ralrimiva 2698	. . 3 |- (F:A-->{B} -> A.x e. A (F` x) = B)
4	1, 3	jca 518	. 2 |- (F:A-->{B} -> (F Fn A /\ A.x e. A (F` x) = B))
5		fneq2 5175	. . . . . . 7 |- (A = (/) -> (F Fn A <-> F Fn (/)))
6		fn0 5203	. . . . . . 7 |- (F Fn (/) <-> F = (/))
7	5, 6	syl6bb 252	. . . . . 6 |- (A = (/) -> (F Fn A <-> F = (/)))
8		f0 5249	. . . . . . 7 |- (/):(/)-->{B}
9		feq1 5211	. . . . . . 7 |- (F = (/) -> (F:(/)-->{B} <-> (/):(/)-->{B}))
10	8, 9	mpbiri 224	. . . . . 6 |- (F = (/) -> F:(/)-->{B})
11	7, 10	syl6bi 219	. . . . 5 |- (A = (/) -> (F Fn A -> F:(/)-->{B}))
12		feq2 5212	. . . . 5 |- (A = (/) -> (F:A-->{B} <-> F:(/)-->{B}))
13	11, 12	sylibrd 225	. . . 4 |- (A = (/) -> (F Fn A -> F:A-->{B}))
14	13	adantrd 454	. . 3 |- (A = (/) -> ((F Fn A /\ A.x e. A (F` x) = B) -> F:A-->{B}))
15		fvelrnb 5366	. . . . . . . . . 10 |- (F Fn A -> (y e. ran F <-> E.z e. A (F` z) = y))
16		fveq2 5329	. . . . . . . . . . . . . . 15 |- (x = z -> (F` x) = (F` z))
17	16	eqeq1d 2361	. . . . . . . . . . . . . 14 |- (x = z -> ((F` x) = B <-> (F` z) = B))
18	17	rspccva 2955	. . . . . . . . . . . . 13 |- ((A.x e. A (F` x) = B /\ z e. A) -> (F` z) = B)
19	18	eqeq1d 2361	. . . . . . . . . . . 12 |- ((A.x e. A (F` x) = B /\ z e. A) -> ((F` z) = y <-> B = y))
20	19	rexbidva 2632	. . . . . . . . . . 11 |- (A.x e. A (F` x) = B -> (E.z e. A (F` z) = y <-> E.z e. A B = y))
21		r19.9rzv 3645	. . . . . . . . . . . 12 |- (A =/= (/) -> (B = y <-> E.z e. A B = y))
22	21	bicomd 192	. . . . . . . . . . 11 |- (A =/= (/) -> (E.z e. A B = y <-> B = y))
23	20, 22	sylan9bbr 681	. . . . . . . . . 10 |- ((A =/= (/) /\ A.x e. A (F` x) = B) -> (E.z e. A (F` z) = y <-> B = y))
24	15, 23	sylan9bbr 681	. . . . . . . . 9 |- (((A =/= (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> (y e. ran F <-> B = y))
25		elsn 3749	. . . . . . . . . 10 |- (y e. {B} <-> y = B)
26		eqcom 2355	. . . . . . . . . 10 |- (y = B <-> B = y)
27	25, 26	bitr2i 241	. . . . . . . . 9 |- (B = y <-> y e. {B})
28	24, 27	syl6bb 252	. . . . . . . 8 |- (((A =/= (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> (y e. ran F <-> y e. {B}))
29	28	eqrdv 2351	. . . . . . 7 |- (((A =/= (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> ran F = {B})
30	29	an32s 779	. . . . . 6 |- (((A =/= (/) /\ F Fn A) /\ A.x e. A (F` x) = B) -> ran F = {B})
31	30	exp31 587	. . . . 5 |- (A =/= (/) -> (F Fn A -> (A.x e. A (F` x) = B -> ran F = {B})))
32	31	imdistand 673	. . . 4 |- (A =/= (/) -> ((F Fn A /\ A.x e. A (F` x) = B) -> (F Fn A /\ ran F = {B})))
33		df-fo 4794	. . . . 5 |- (F:A-onto->{B} <-> (F Fn A /\ ran F = {B}))
34		fof 5270	. . . . 5 |- (F:A-onto->{B} -> F:A-->{B})
35	33, 34	sylbir 204	. . . 4 |- ((F Fn A /\ ran F = {B}) -> F:A-->{B})
36	32, 35	syl6 29	. . 3 |- (A =/= (/) -> ((F Fn A /\ A.x e. A (F` x) = B) -> F:A-->{B}))
37	14, 36	pm2.61ine 2593	. 2 |- ((F Fn A /\ A.x e. A (F` x) = B) -> F:A-->{B})
38	4, 37	impbii 180	1 |- (F:A-->{B} <-> (F Fn A /\ A.x e. A (F` x) = B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  A.wral 2615  E.wrex 2616  (/)c0 3551  {csn 3738  ran crn 4774   Fn wfn 4777  -->wf 4778  -onto->wfo 4780  ` cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796

This theorem is referenced by:  fconst3  5458