Theorem fconst5 3848

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 3472	. . . . 5 |- (A =/= (/) -> ran ( A X. {B}) = {B})
2	1	eqeq2d 1486	. . . 4 |- (A =/= (/) -> (ran F = ran ( A X. {B}) <-> ran F = {B}))
3		rneq 3339	. . . 4 |- (F = (A X. {B}) -> ran F = ran ( A X. {B}))
4	2, 3	syl5bi 208	. . 3 |- (A =/= (/) -> (F = (A X. {B}) -> ran F = {B}))
5	4	adantl 388	. 2 |- ((F Fn A /\ A =/= (/)) -> (F = (A X. {B}) -> ran F = {B}))
6		fconst2g 3845	. . . . . 6 |- (B e. V -> (F:A-->{B} <-> F = (A X. {B})))
7		df-fo 3196	. . . . . . 7 |- (F:A-onto->{B} <-> (F Fn A /\ ran F = {B}))
8		fof 3672	. . . . . . 7 |- (F:A-onto->{B} -> F:A-->{B})
9	7, 8	sylbir 201	. . . . . 6 |- ((F Fn A /\ ran F = {B}) -> F:A-->{B})
10	6, 9	syl5bi 208	. . . . 5 |- (B e. V -> ((F Fn A /\ ran F = {B}) -> F = (A X. {B})))
11	10	exp3a 375	. . . 4 |- (B e. V -> (F Fn A -> (ran F = {B} -> F = (A X. {B}))))
12	11	adantrd 391	. . 3 |- (B e. V -> ((F Fn A /\ A =/= (/)) -> (ran F = {B} -> F = (A X. {B}))))
13		snprc 2443	. . . . . 6 |- (-. B e. V <-> {B} = (/))
14		relrn0 3356	. . . . . . . . . 10 |- (Rel F -> (F = (/) <-> ran F = (/)))
15	14	biimprd 154	. . . . . . . . 9 |- (Rel F -> (ran F = (/) -> F = (/)))
16	15	adantl 388	. . . . . . . 8 |- (({B} = (/) /\ Rel F) -> (ran F = (/) -> F = (/)))
17		eqeq2 1484	. . . . . . . . 9 |- ({B} = (/) -> (ran F = {B} <-> ran F = (/)))
18	17	adantr 389	. . . . . . . 8 |- (({B} = (/) /\ Rel F) -> (ran F = {B} <-> ran F = (/)))
19		xpeq2 3201	. . . . . . . . . . 11 |- ({B} = (/) -> (A X. {B}) = (A X. (/)))
20		xp0 3465	. . . . . . . . . . 11 |- (A X. (/)) = (/)
21	19, 20	syl6eq 1523	. . . . . . . . . 10 |- ({B} = (/) -> (A X. {B}) = (/))
22	21	eqeq2d 1486	. . . . . . . . 9 |- ({B} = (/) -> (F = (A X. {B}) <-> F = (/)))
23	22	adantr 389	. . . . . . . 8 |- (({B} = (/) /\ Rel F) -> (F = (A X. {B}) <-> F = (/)))
24	16, 18, 23	3imtr4d 543	. . . . . . 7 |- (({B} = (/) /\ Rel F) -> (ran F = {B} -> F = (A X. {B})))
25	24	ex 373	. . . . . 6 |- ({B} = (/) -> (Rel F -> (ran F = {B} -> F = (A X. {B}))))
26	13, 25	sylbi 199	. . . . 5 |- (-. B e. V -> (Rel F -> (ran F = {B} -> F = (A X. {B}))))
27		fnrel 3586	. . . . 5 |- (F Fn A -> Rel F)
28	26, 27	syl5 21	. . . 4 |- (-. B e. V -> (F Fn A -> (ran F = {B} -> F = (A X. {B}))))
29	28	adantrd 391	. . 3 |- (-. B e. V -> ((F Fn A /\ A =/= (/)) -> (ran F = {B} -> F = (A X. {B}))))
30	12, 29	pm2.61i 126	. 2 |- ((F Fn A /\ A =/= (/)) -> (ran F = {B} -> F = (A X. {B})))
31	5, 30	impbid 516	1 |- ((F Fn A /\ A =/= (/)) -> (F = (A X. {B}) <-> ran F = {B}))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585  Vcvv 1811  (/)c0 2280  {csn 2409   X. cxp 3168  ran crn 3171  Rel wrel 3175   Fn wfn 3177  -->wf 3178  -onto->wfo 3180

This theorem is referenced by:  nvo00 8424

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-9 965  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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