Theorem f1o00 5410

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 5383	. 2 |- ( F : (/) -1-1-onto-> A <-> ( F Fn (/) /\ `' F Fn A ) )
2		fn0 5250	. . . . . 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 4761	. . . . 5 |- dom (/) = (/)
6		cnveq 4721	. . . . . . . . . 10 |- ( F = (/) -> `' F = `' (/) )
7		cnv0 4950	. . . . . . . . . 10 |- `' (/) = (/)
8	6, 7	eqtrdi 2189	. . . . . . . . 9 |- ( F = (/) -> `' F = (/) )
9	2, 8	sylbi 120	. . . . . . . 8 |- ( F Fn (/) -> `' F = (/) )
10	9	fneq1d 5221	. . . . . . 7 |- ( F Fn (/) -> ( `' F Fn A <-> (/) Fn A ) )
11	10	biimpa 294	. . . . . 6 |- ( ( F Fn (/) /\ `' F Fn A ) -> (/) Fn A )
12		fndm 5230	. . . . . 6 |- ( (/) Fn A -> dom (/) = A )
13	11, 12	syl 14	. . . . 5 |- ( ( F Fn (/) /\ `' F Fn A ) -> dom (/) = A )
14	5, 13	syl5reqr 2188	. . . 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 2140	. . . . . 6 |- (/) = (/)
19		fn0 5250	. . . . . 6 |- ( (/) Fn (/) <-> (/) = (/) )
20	18, 19	mpbir 145	. . . . 5 |- (/) Fn (/)
21	8	fneq1d 5221	. . . . . 6 |- ( F = (/) -> ( `' F Fn A <-> (/) Fn A ) )
22		fneq2 5220	. . . . . 6 |- ( A = (/) -> ( (/) Fn A <-> (/) Fn (/) ) )
23	21, 22	sylan9bb 458	. . . . 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 1332   (/)c0 3368   `'ccnv 4546   dom cdm 4547   Fn wfn 5126   -1-1-onto->wf1o 5130

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138

This theorem is referenced by:  fo00  5411  f1o0  5412  en0  6697