Theorem fconst5 3787

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 3421	. . . . 5 |- (A =/= (/) -> ran ( A X. {B}) = {B})
2	1	eqeq2d 1462	. . . 4 |- (A =/= (/) -> (ran F = ran ( A X. {B}) <-> ran F = {B}))
3		rneq 3298	. . . 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 3784	. . . . . 6 |- (B e. V -> (F:A-->{B} <-> F = (A X. {B})))
7		df-fo 3159	. . . . . . 7 |- (F:A-onto->{B} <-> (F Fn A /\ ran F = {B}))
8		fof 3611	. . . . . . 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 2414	. . . . . 6 |- (-. B e. V <-> {B} = (/))
14		relrn0 3312	. . . . . . . . . 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 1460	. . . . . . . . 9 |- ({B} = (/) -> (ran F = {B} <-> ran F = (/)))
18	17	adantr 389	. . . . . . . 8 |- (({B} = (/) /\ Rel F) -> (ran F = {B} <-> ran F = (/)))
19		xpeq2 3164	. . . . . . . . . . 11 |- ({B} = (/) -> (A X. {B}) = (A X. (/)))
20		xp0 3414	. . . . . . . . . . 11 |- (A X. (/)) = (/)
21	19, 20	syl6eq 1499	. . . . . . . . . 10 |- ({B} = (/) -> (A X. {B}) = (/))
22	21	eqeq2d 1462	. . . . . . . . 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 541	. . . . . . 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 3526	. . . . 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 514	1 |- ((F Fn A /\ A =/= (/)) -> (F = (A X. {B}) <-> ran F = {B}))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 1099   e. wcel 1105   =/= wne 1561  Vcvv 1786  (/)c0 2251  {csn 2380   X. cxp 3131  ran crn 3134  Rel wrel 3138   Fn wfn 3140  -->wf 3141  -onto->wfo 3143

This theorem is referenced by:  nvo00 8291

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-fo 3159  df-fv 3161

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