Theorem f1o00 4668

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		dff1o4 4644	. 2 |- (F:(/)-1-1-onto->A <-> (F Fn (/) /\ `'F Fn A))
2		fn0 4532	. . . . . 6 |- (F Fn (/) <-> F = (/))
3	2	biimpi 168	. . . . 5 |- (F Fn (/) -> F = (/))
4	3	adantr 425	. . . 4 |- ((F Fn (/) /\ `'F Fn A) -> F = (/))
5		cnveq 4135	. . . . . . . . . 10 |- (F = (/) -> `'F = `'(/))
6		cnv0 4319	. . . . . . . . . 10 |- `'(/) = (/)
7	5, 6	syl6eq 1944	. . . . . . . . 9 |- (F = (/) -> `'F = (/))
8	2, 7	sylbi 216	. . . . . . . 8 |- (F Fn (/) -> `'F = (/))
9	8	fneq1d 4505	. . . . . . 7 |- (F Fn (/) -> (`'F Fn A <-> (/) Fn A))
10	9	biimpa 460	. . . . . 6 |- ((F Fn (/) /\ `'F Fn A) -> (/) Fn A)
11		fndm 4512	. . . . . 6 |- ((/) Fn A -> dom (/) = A)
12	10, 11	syl 12	. . . . 5 |- ((F Fn (/) /\ `'F Fn A) -> dom (/) = A)
13		dm0 4170	. . . . 5 |- dom (/) = (/)
14	12, 13	syl5reqr 1943	. . . 4 |- ((F Fn (/) /\ `'F Fn A) -> A = (/))
15	4, 14	jca 310	. . 3 |- ((F Fn (/) /\ `'F Fn A) -> (F = (/) /\ A = (/)))
16	2	biimpri 169	. . . . 5 |- (F = (/) -> F Fn (/))
17	16	adantr 425	. . . 4 |- ((F = (/) /\ A = (/)) -> F Fn (/))
18		eqid 1884	. . . . . 6 |- (/) = (/)
19		fn0 4532	. . . . . 6 |- ((/) Fn (/) <-> (/) = (/))
20	18, 19	mpbir 207	. . . . 5 |- (/) Fn (/)
21	7	fneq1d 4505	. . . . . 6 |- (F = (/) -> (`'F Fn A <-> (/) Fn A))
22		fneq2 4504	. . . . . 6 |- (A = (/) -> ((/) Fn A <-> (/) Fn (/)))
23	21, 22	sylan9bb 599	. . . . 5 |- ((F = (/) /\ A = (/)) -> (`'F Fn A <-> (/) Fn (/)))
24	20, 23	mpbiri 211	. . . 4 |- ((F = (/) /\ A = (/)) -> `'F Fn A)
25	17, 24	jca 310	. . 3 |- ((F = (/) /\ A = (/)) -> (F Fn (/) /\ `'F Fn A))
26	15, 25	impbii 174	. 2 |- ((F Fn (/) /\ `'F Fn A) <-> (F = (/) /\ A = (/)))
27	1, 26	bitri 190	1 |- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298  (/)c0 2875  `'ccnv 3985  dom cdm 3986   Fn wfn 3993  -1-1-onto->wf1o 3997

This theorem is referenced by:  fo00 4669  f1o0 4670  en0 5482  ac6sfi 5509  fbssint 10279

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  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-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-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-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

Copyright terms: Public domain