Theorem f1o00 5181

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 5154	. 2 |- ( F : (/) -1-1-onto-> A <-> ( F Fn (/) /\ `' F Fn A ) )
2		fn0 5038	. . . . . 6 |- ( F Fn (/) <-> F = (/) )
3	2	biimpi 118	. . . . 5 |- ( F Fn (/) -> F = (/) )
4	3	adantr 270	. . . 4 |- ( ( F Fn (/) /\ `' F Fn A ) -> F = (/) )
5		dm0 4567	. . . . 5 |- dom (/) = (/)
6		cnveq 4527	. . . . . . . . . 10 |- ( F = (/) -> `' F = `' (/) )
7		cnv0 4747	. . . . . . . . . 10 |- `' (/) = (/)
8	6, 7	syl6eq 2129	. . . . . . . . 9 |- ( F = (/) -> `' F = (/) )
9	2, 8	sylbi 119	. . . . . . . 8 |- ( F Fn (/) -> `' F = (/) )
10	9	fneq1d 5009	. . . . . . 7 |- ( F Fn (/) -> ( `' F Fn A <-> (/) Fn A ) )
11	10	biimpa 290	. . . . . 6 |- ( ( F Fn (/) /\ `' F Fn A ) -> (/) Fn A )
12		fndm 5018	. . . . . 6 |- ( (/) Fn A -> dom (/) = A )
13	11, 12	syl 14	. . . . 5 |- ( ( F Fn (/) /\ `' F Fn A ) -> dom (/) = A )
14	5, 13	syl5reqr 2128	. . . 4 |- ( ( F Fn (/) /\ `' F Fn A ) -> A = (/) )
15	4, 14	jca 300	. . 3 |- ( ( F Fn (/) /\ `' F Fn A ) -> ( F = (/) /\ A = (/) ) )
16	2	biimpri 131	. . . . 5 |- ( F = (/) -> F Fn (/) )
17	16	adantr 270	. . . 4 |- ( ( F = (/) /\ A = (/) ) -> F Fn (/) )
18		eqid 2081	. . . . . 6 |- (/) = (/)
19		fn0 5038	. . . . . 6 |- ( (/) Fn (/) <-> (/) = (/) )
20	18, 19	mpbir 144	. . . . 5 |- (/) Fn (/)
21	8	fneq1d 5009	. . . . . 6 |- ( F = (/) -> ( `' F Fn A <-> (/) Fn A ) )
22		fneq2 5008	. . . . . 6 |- ( A = (/) -> ( (/) Fn A <-> (/) Fn (/) ) )
23	21, 22	sylan9bb 449	. . . . 5 |- ( ( F = (/) /\ A = (/) ) -> ( `' F Fn A <-> (/) Fn (/) ) )
24	20, 23	mpbiri 166	. . . 4 |- ( ( F = (/) /\ A = (/) ) -> `' F Fn A )
25	17, 24	jca 300	. . 3 |- ( ( F = (/) /\ A = (/) ) -> ( F Fn (/) /\ `' F Fn A ) )
26	15, 25	impbii 124	. 2 |- ( ( F Fn (/) /\ `' F Fn A ) <-> ( F = (/) /\ A = (/) ) )
27	1, 26	bitri 182	1 |- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   (/)c0 3251   `'ccnv 4362   dom cdm 4363   Fn wfn 4917   -1-1-onto->wf1o 4921

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929

This theorem is referenced by:  fo00  5182  f1o0  5183  en0  6298