Theorem fconstfv 3849

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 3847.

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

Step	Hyp	Ref	Expression
1		ffn 3627	. . 3 |- (F:A-->{B} -> F Fn A)
2		fvconst 3839	. . . 4 |- ((F:A-->{B} /\ x e. A) -> (F` x) = B)
3	2	r19.21aiva 1714	. . 3 |- (F:A-->{B} -> A.x e. A (F` x) = B)
4	1, 3	jca 288	. 2 |- (F:A-->{B} -> (F Fn A /\ A.x e. A (F` x) = B))
5		fneq2 3583	. . . . . . 7 |- (A = (/) -> (F Fn A <-> F Fn (/)))
6		fn0 3605	. . . . . . 7 |- (F Fn (/) <-> F = (/))
7	5, 6	syl6bb 536	. . . . . 6 |- (A = (/) -> (F Fn A <-> F = (/)))
8		f0 3656	. . . . . . 7 |- (/):(/)-->{B}
9		feq1 3620	. . . . . . 7 |- (F = (/) -> (F:(/)-->{B} <-> (/):(/)-->{B}))
10	8, 9	mpbiri 194	. . . . . 6 |- (F = (/) -> F:(/)-->{B})
11	7, 10	syl6bi 214	. . . . 5 |- (A = (/) -> (F Fn A -> F:(/)-->{B}))
12		feq2 3621	. . . . 5 |- (A = (/) -> (F:A-->{B} <-> F:(/)-->{B}))
13	11, 12	sylibrd 204	. . . 4 |- (A = (/) -> (F Fn A -> F:A-->{B}))
14	13	adantrd 391	. . 3 |- (A = (/) -> ((F Fn A /\ A.x e. A (F` x) = B) -> F:A-->{B}))
15		fvelrnb 3760	. . . . . . . . . 10 |- (F Fn A -> (y e. ran F <-> E.z e. A (F` z) = y))
16		fveq2 3724	. . . . . . . . . . . . . . 15 |- (x = z -> (F` x) = (F` z))
17	16	eqeq1d 1483	. . . . . . . . . . . . . 14 |- (x = z -> ((F` x) = B <-> (F` z) = B))
18	17	rcla4cva 1876	. . . . . . . . . . . . 13 |- ((A.x e. A (F` x) = B /\ z e. A) -> (F` z) = B)
19	18	eqeq1d 1483	. . . . . . . . . . . 12 |- ((A.x e. A (F` x) = B /\ z e. A) -> ((F` z) = y <-> B = y))
20	19	rexbidva 1660	. . . . . . . . . . 11 |- (A.x e. A (F` x) = B -> (E.z e. A (F` z) = y <-> E.z e. A B = y))
21		r19.9rzv 2349	. . . . . . . . . . . 12 |- (A =/= (/) -> (B = y <-> E.z e. A B = y))
22	21	bicomd 521	. . . . . . . . . . 11 |- (A =/= (/) -> (E.z e. A B = y <-> B = y))
23	20, 22	sylan9bbr 541	. . . . . . . . . 10 |- ((A =/= (/) /\ A.x e. A (F` x) = B) -> (E.z e. A (F` z) = y <-> B = y))
24	15, 23	sylan9bbr 541	. . . . . . . . 9 |- (((A =/= (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> (y e. ran F <-> B = y))
25		elsn 2421	. . . . . . . . . 10 |- (y e. {B} <-> y = B)
26		eqcom 1477	. . . . . . . . . 10 |- (y = B <-> B = y)
27	25, 26	bitr2 174	. . . . . . . . 9 |- (B = y <-> y e. {B})
28	24, 27	syl6bb 536	. . . . . . . 8 |- (((A =/= (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> (y e. ran F <-> y e. {B}))
29	28	eqrdv 1473	. . . . . . 7 |- (((A =/= (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> ran F = {B})
30	29	an1rs 489	. . . . . 6 |- (((A =/= (/) /\ F Fn A) /\ A.x e. A (F` x) = B) -> ran F = {B})
31	30	exp31 376	. . . . 5 |- (A =/= (/) -> (F Fn A -> (A.x e. A (F` x) = B -> ran F = {B})))
32	31	imdistand 445	. . . 4 |- (A =/= (/) -> ((F Fn A /\ A.x e. A (F` x) = B) -> (F Fn A /\ ran F = {B})))
33		df-fo 3196	. . . . 5 |- (F:A-onto->{B} <-> (F Fn A /\ ran F = {B}))
34		fof 3672	. . . . 5 |- (F:A-onto->{B} -> F:A-->{B})
35	33, 34	sylbir 201	. . . 4 |- ((F Fn A /\ ran F = {B}) -> F:A-->{B})
36	32, 35	syl6 22	. . 3 |- (A =/= (/) -> ((F Fn A /\ A.x e. A (F` x) = B) -> F:A-->{B}))
37	14, 36	pm2.61ine 1634	. 2 |- ((F Fn A /\ A.x e. A (F` x) = B) -> F:A-->{B})
38	4, 37	impbi 157	1 |- (F:A-->{B} <-> (F Fn A /\ A.x e. A (F` x) = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646  (/)c0 2280  {csn 2409  ran crn 3171   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  ` cfv 3182

This theorem is referenced by:  fconst3 3850  lnon0 8458  df0op2 9678

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-nul 2710  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-ral 1649  df-rex 1650  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-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198

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