Theorem fconstfv 3840

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 3838.

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 3619	. . 3 |- (F:A-->{B} -> F Fn A)
2		fvconst 3830	. . . 4 |- ((F:A-->{B} /\ x e. A) -> (F` x) = B)
3	2	r19.21aiva 1711	. . 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 3575	. . . . . . 7 |- (A = (/) -> (F Fn A <-> F Fn (/)))
6		fn0 3597	. . . . . . 7 |- (F Fn (/) <-> F = (/))
7	5, 6	syl6bb 535	. . . . . 6 |- (A = (/) -> (F Fn A <-> F = (/)))
8		f0 3647	. . . . . . 7 |- (/):(/)-->{B}
9		feq1 3612	. . . . . . 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 3613	. . . . 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 3751	. . . . . . . . . 10 |- (F Fn A -> (y e. ran F <-> E.z e. A (F` z) = y))
16		fveq2 3715	. . . . . . . . . . . . . . 15 |- (x = z -> (F` x) = (F` z))
17	16	eqeq1d 1480	. . . . . . . . . . . . . 14 |- (x = z -> ((F` x) = B <-> (F` z) = B))
18	17	rcla4cva 1872	. . . . . . . . . . . . 13 |- ((A.x e. A (F` x) = B /\ z e. A) -> (F` z) = B)
19	18	eqeq1d 1480	. . . . . . . . . . . 12 |- ((A.x e. A (F` x) = B /\ z e. A) -> ((F` z) = y <-> B = y))
20	19	rexbidva 1657	. . . . . . . . . . 11 |- (A.x e. A (F` x) = B -> (E.z e. A (F` z) = y <-> E.z e. A B = y))
21		r19.9rzv 2345	. . . . . . . . . . . 12 |- (A =/= (/) -> (B = y <-> E.z e. A B = y))
22	21	bicomd 520	. . . . . . . . . . 11 |- (A =/= (/) -> (E.z e. A B = y <-> B = y))
23	20, 22	sylan9bbr 540	. . . . . . . . . 10 |- ((A =/= (/) /\ A.x e. A (F` x) = B) -> (E.z e. A (F` z) = y <-> B = y))
24	15, 23	sylan9bbr 540	. . . . . . . . 9 |- (((A =/= (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> (y e. ran F <-> B = y))
25		elsn 2417	. . . . . . . . . 10 |- (y e. {B} <-> y = B)
26		eqcom 1474	. . . . . . . . . 10 |- (y = B <-> B = y)
27	25, 26	bitr2 174	. . . . . . . . 9 |- (B = y <-> y e. {B})
28	24, 27	syl6bb 535	. . . . . . . 8 |- (((A =/= (/) /\ A.x e. A (F` x) = B) /\ F Fn A) -> (y e. ran F <-> y e. {B}))
29	28	eqrdv 1471	. . . . . . 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 3191	. . . . 5 |- (F:A-onto->{B} <-> (F Fn A /\ ran F = {B}))
34		fof 3663	. . . . 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 1631	. 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 954   e. wcel 956   =/= wne 1582  A.wral 1642  E.wrex 1643  (/)c0 2276  {csn 2405  ran crn 3166   Fn wfn 3172  -->wf 3173  -onto->wfo 3175  ` cfv 3177

This theorem is referenced by:  fconst3 3841  df0op2 9618

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-nul 2705  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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