Theorem f1o00 5402

Description: One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)

Assertion

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

Proof of Theorem f1o00

Step	Hyp	Ref	Expression
1		dff1o4 5375	. 2 |- ( F : (/) -1-1-onto-> A <-> ( F Fn (/) /\ `' F Fn A ) )
2		fn0 5242	. . . . . 6 |- ( F Fn (/) <-> F = (/) )
3	2	biimpi 119	. . . . 5 |- ( F Fn (/) -> F = (/) )
4	3	adantr 274	. . . 4 |- ( ( F Fn (/) /\ `' F Fn A ) -> F = (/) )
5		dm0 4753	. . . . 5 |- dom (/) = (/)
6		cnveq 4713	. . . . . . . . . 10 |- ( F = (/) -> `' F = `' (/) )
7		cnv0 4942	. . . . . . . . . 10 |- `' (/) = (/)
8	6, 7	syl6eq 2188	. . . . . . . . 9 |- ( F = (/) -> `' F = (/) )
9	2, 8	sylbi 120	. . . . . . . 8 |- ( F Fn (/) -> `' F = (/) )
10	9	fneq1d 5213	. . . . . . 7 |- ( F Fn (/) -> ( `' F Fn A <-> (/) Fn A ) )
11	10	biimpa 294	. . . . . 6 |- ( ( F Fn (/) /\ `' F Fn A ) -> (/) Fn A )
12		fndm 5222	. . . . . 6 |- ( (/) Fn A -> dom (/) = A )
13	11, 12	syl 14	. . . . 5 |- ( ( F Fn (/) /\ `' F Fn A ) -> dom (/) = A )
14	5, 13	syl5reqr 2187	. . . 4 |- ( ( F Fn (/) /\ `' F Fn A ) -> A = (/) )
15	4, 14	jca 304	. . 3 |- ( ( F Fn (/) /\ `' F Fn A ) -> ( F = (/) /\ A = (/) ) )
16	2	biimpri 132	. . . . 5 |- ( F = (/) -> F Fn (/) )
17	16	adantr 274	. . . 4 |- ( ( F = (/) /\ A = (/) ) -> F Fn (/) )
18		eqid 2139	. . . . . 6 |- (/) = (/)
19		fn0 5242	. . . . . 6 |- ( (/) Fn (/) <-> (/) = (/) )
20	18, 19	mpbir 145	. . . . 5 |- (/) Fn (/)
21	8	fneq1d 5213	. . . . . 6 |- ( F = (/) -> ( `' F Fn A <-> (/) Fn A ) )
22		fneq2 5212	. . . . . 6 |- ( A = (/) -> ( (/) Fn A <-> (/) Fn (/) ) )
23	21, 22	sylan9bb 457	. . . . 5 |- ( ( F = (/) /\ A = (/) ) -> ( `' F Fn A <-> (/) Fn (/) ) )
24	20, 23	mpbiri 167	. . . 4 |- ( ( F = (/) /\ A = (/) ) -> `' F Fn A )
25	17, 24	jca 304	. . 3 |- ( ( F = (/) /\ A = (/) ) -> ( F Fn (/) /\ `' F Fn A ) )
26	15, 25	impbii 125	. 2 |- ( ( F Fn (/) /\ `' F Fn A ) <-> ( F = (/) /\ A = (/) ) )
27	1, 26	bitri 183	1 |- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   (/)c0 3363   `'ccnv 4538   dom cdm 4539   Fn wfn 5118   -1-1-onto->wf1o 5122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130

This theorem is referenced by:  fo00  5403  f1o0  5404  en0  6689