Theorem fconst5 4824

Description: Two ways to express that a function is constant.

Assertion

Ref	Expression
fconst5	|- ((F Fn A /\ A =/= (/)) -> (F = (A X. {B}) <-> ran F = {B}))

Proof of Theorem fconst5

Step	Hyp	Ref	Expression
1		rnxp 4342	. . . . 5 |- (A =/= (/) -> ran ( A X. {B}) = {B})
2	1	eqeq2d 1895	. . . 4 |- (A =/= (/) -> (ran F = ran ( A X. {B}) <-> ran F = {B}))
3		rneq 4186	. . . 4 |- (F = (A X. {B}) -> ran F = ran ( A X. {B}))
4	2, 3	syl5bi 225	. . 3 |- (A =/= (/) -> (F = (A X. {B}) -> ran F = {B}))
5	4	adantl 424	. 2 |- ((F Fn A /\ A =/= (/)) -> (F = (A X. {B}) -> ran F = {B}))
6		fconst2g 4821	. . . . . 6 |- (B e. _V -> (F:A-->{B} <-> F = (A X. {B})))
7		df-fo 4012	. . . . . . 7 |- (F:A-onto->{B} <-> (F Fn A /\ ran F = {B}))
8		fof 4617	. . . . . . 7 |- (F:A-onto->{B} -> F:A-->{B})
9	7, 8	sylbir 218	. . . . . 6 |- ((F Fn A /\ ran F = {B}) -> F:A-->{B})
10	6, 9	syl5bi 225	. . . . 5 |- (B e. _V -> ((F Fn A /\ ran F = {B}) -> F = (A X. {B})))
11	10	exp3a 405	. . . 4 |- (B e. _V -> (F Fn A -> (ran F = {B} -> F = (A X. {B}))))
12	11	adantrd 427	. . 3 |- (B e. _V -> ((F Fn A /\ A =/= (/)) -> (ran F = {B} -> F = (A X. {B}))))
13		snprc 3092	. . . . . 6 |- (-. B e. _V <-> {B} = (/))
14		relrn0 4204	. . . . . . . . . 10 |- (Rel F -> (F = (/) <-> ran F = (/)))
15	14	biimprd 171	. . . . . . . . 9 |- (Rel F -> (ran F = (/) -> F = (/)))
16	15	adantl 424	. . . . . . . 8 |- (({B} = (/) /\ Rel F) -> (ran F = (/) -> F = (/)))
17		eqeq2 1893	. . . . . . . . 9 |- ({B} = (/) -> (ran F = {B} <-> ran F = (/)))
18	17	adantr 425	. . . . . . . 8 |- (({B} = (/) /\ Rel F) -> (ran F = {B} <-> ran F = (/)))
19		xpeq2 4017	. . . . . . . . . . 11 |- ({B} = (/) -> (A X. {B}) = (A X. (/)))
20		xp0 4334	. . . . . . . . . . 11 |- (A X. (/)) = (/)
21	19, 20	syl6eq 1944	. . . . . . . . . 10 |- ({B} = (/) -> (A X. {B}) = (/))
22	21	eqeq2d 1895	. . . . . . . . 9 |- ({B} = (/) -> (F = (A X. {B}) <-> F = (/)))
23	22	adantr 425	. . . . . . . 8 |- (({B} = (/) /\ Rel F) -> (F = (A X. {B}) <-> F = (/)))
24	16, 18, 23	3imtr4d 602	. . . . . . 7 |- (({B} = (/) /\ Rel F) -> (ran F = {B} -> F = (A X. {B})))
25	24	ex 402	. . . . . 6 |- ({B} = (/) -> (Rel F -> (ran F = {B} -> F = (A X. {B}))))
26	13, 25	sylbi 216	. . . . 5 |- (-. B e. _V -> (Rel F -> (ran F = {B} -> F = (A X. {B}))))
27		fnrel 4511	. . . . 5 |- (F Fn A -> Rel F)
28	26, 27	syl5 20	. . . 4 |- (-. B e. _V -> (F Fn A -> (ran F = {B} -> F = (A X. {B}))))
29	28	adantrd 427	. . 3 |- (-. B e. _V -> ((F Fn A /\ A =/= (/)) -> (ran F = {B} -> F = (A X. {B}))))
30	12, 29	pm2.61i 140	. 2 |- ((F Fn A /\ A =/= (/)) -> (ran F = {B} -> F = (A X. {B})))
31	5, 30	impbid 574	1 |- ((F Fn A /\ A =/= (/)) -> (F = (A X. {B}) <-> ran F = {B}))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  (/)c0 2875  {csn 3044   X. cxp 3984  ran crn 3987  Rel wrel 3991   Fn wfn 3993  -->wf 3994  -onto->wfo 3996

This theorem is referenced by:  nvo00 9763

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014

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