Theorem f1o00 3699

Description: One-to-one onto mapping of the empty set.

Assertion

Ref	Expression
f1o00	|- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))

Proof of Theorem f1o00

Step	Hyp	Ref	Expression
1		f1o4 3681	. 2 |- (F:(/)-1-1-onto->A <-> (F Fn (/) /\ `'F Fn A))
2		fn0 3591	. . . . . 6 |- (F Fn (/) <-> F = (/))
3	2	biimp 151	. . . . 5 |- (F Fn (/) -> F = (/))
4	3	adantr 389	. . . 4 |- ((F Fn (/) /\ `'F Fn A) -> F = (/))
5		cnveq 3281	. . . . . . . . . 10 |- (F = (/) -> `'F = `'(/))
6		cnv0 3432	. . . . . . . . . 10 |- `'(/) = (/)
7	5, 6	syl6eq 1515	. . . . . . . . 9 |- (F = (/) -> `'F = (/))
8	2, 7	sylbi 199	. . . . . . . 8 |- (F Fn (/) -> `'F = (/))
9		fneq1 3568	. . . . . . . 8 |- (`'F = (/) -> (`'F Fn A <-> (/) Fn A))
10	8, 9	syl 10	. . . . . . 7 |- (F Fn (/) -> (`'F Fn A <-> (/) Fn A))
11	10	biimpa 416	. . . . . 6 |- ((F Fn (/) /\ `'F Fn A) -> (/) Fn A)
12		fndm 3573	. . . . . 6 |- ((/) Fn A -> dom (/) = A)
13	11, 12	syl 10	. . . . 5 |- ((F Fn (/) /\ `'F Fn A) -> dom (/) = A)
14		dm0 3312	. . . . 5 |- dom (/) = (/)
15	13, 14	syl5reqr 1514	. . . 4 |- ((F Fn (/) /\ `'F Fn A) -> A = (/))
16	4, 15	jca 288	. . 3 |- ((F Fn (/) /\ `'F Fn A) -> (F = (/) /\ A = (/)))
17	2	biimpr 152	. . . . 5 |- (F = (/) -> F Fn (/))
18	17	adantr 389	. . . 4 |- ((F = (/) /\ A = (/)) -> F Fn (/))
19		eqid 1468	. . . . . 6 |- (/) = (/)
20		fn0 3591	. . . . . 6 |- ((/) Fn (/) <-> (/) = (/))
21	19, 20	mpbir 190	. . . . 5 |- (/) Fn (/)
22	7, 9	syl 10	. . . . . 6 |- (F = (/) -> (`'F Fn A <-> (/) Fn A))
23		fneq2 3569	. . . . . 6 |- (A = (/) -> ((/) Fn A <-> (/) Fn (/)))
24	22, 23	sylan9bb 538	. . . . 5 |- ((F = (/) /\ A = (/)) -> (`'F Fn A <-> (/) Fn (/)))
25	21, 24	mpbiri 194	. . . 4 |- ((F = (/) /\ A = (/)) -> `'F Fn A)
26	18, 25	jca 288	. . 3 |- ((F = (/) /\ A = (/)) -> (F Fn (/) /\ `'F Fn A))
27	16, 26	impbi 157	. 2 |- ((F Fn (/) /\ `'F Fn A) <-> (F = (/) /\ A = (/)))
28	1, 27	bitr 173	1 |- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953  (/)c0 2270  `'ccnv 3159  dom cdm 3160   Fn wfn 3167  -1-1-onto->wf1o 3171

This theorem is referenced by:  fo00 3700  f1o0 3701  en0 4404

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187

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