Theorem f1o00 5616

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 5588	. 2 |- ( F : (/) -1-1-onto-> A <-> ( F Fn (/) /\ `' F Fn A ) )
2		fn0 5449	. . . . . 6 |- ( F Fn (/) <-> F = (/) )
3	2	biimpi 120	. . . . 5 |- ( F Fn (/) -> F = (/) )
4	3	adantr 276	. . . 4 |- ( ( F Fn (/) /\ `' F Fn A ) -> F = (/) )
5		cnveq 4902	. . . . . . . . . 10 |- ( F = (/) -> `' F = `' (/) )
6		cnv0 5138	. . . . . . . . . 10 |- `' (/) = (/)
7	5, 6	eqtrdi 2278	. . . . . . . . 9 |- ( F = (/) -> `' F = (/) )
8	2, 7	sylbi 121	. . . . . . . 8 |- ( F Fn (/) -> `' F = (/) )
9	8	fneq1d 5417	. . . . . . 7 |- ( F Fn (/) -> ( `' F Fn A <-> (/) Fn A ) )
10	9	biimpa 296	. . . . . 6 |- ( ( F Fn (/) /\ `' F Fn A ) -> (/) Fn A )
11		fndm 5426	. . . . . 6 |- ( (/) Fn A -> dom (/) = A )
12	10, 11	syl 14	. . . . 5 |- ( ( F Fn (/) /\ `' F Fn A ) -> dom (/) = A )
13		dm0 4943	. . . . 5 |- dom (/) = (/)
14	12, 13	eqtr3di 2277	. . . 4 |- ( ( F Fn (/) /\ `' F Fn A ) -> A = (/) )
15	4, 14	jca 306	. . 3 |- ( ( F Fn (/) /\ `' F Fn A ) -> ( F = (/) /\ A = (/) ) )
16	2	biimpri 133	. . . . 5 |- ( F = (/) -> F Fn (/) )
17	16	adantr 276	. . . 4 |- ( ( F = (/) /\ A = (/) ) -> F Fn (/) )
18		eqid 2229	. . . . . 6 |- (/) = (/)
19		fn0 5449	. . . . . 6 |- ( (/) Fn (/) <-> (/) = (/) )
20	18, 19	mpbir 146	. . . . 5 |- (/) Fn (/)
21	7	fneq1d 5417	. . . . . 6 |- ( F = (/) -> ( `' F Fn A <-> (/) Fn A ) )
22		fneq2 5416	. . . . . 6 |- ( A = (/) -> ( (/) Fn A <-> (/) Fn (/) ) )
23	21, 22	sylan9bb 462	. . . . 5 |- ( ( F = (/) /\ A = (/) ) -> ( `' F Fn A <-> (/) Fn (/) ) )
24	20, 23	mpbiri 168	. . . 4 |- ( ( F = (/) /\ A = (/) ) -> `' F Fn A )
25	17, 24	jca 306	. . 3 |- ( ( F = (/) /\ A = (/) ) -> ( F Fn (/) /\ `' F Fn A ) )
26	15, 25	impbii 126	. 2 |- ( ( F Fn (/) /\ `' F Fn A ) <-> ( F = (/) /\ A = (/) ) )
27	1, 26	bitri 184	1 |- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   (/)c0 3492   `'ccnv 4722   dom cdm 4723   Fn wfn 5319   -1-1-onto->wf1o 5323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331

This theorem is referenced by:  fo00  5617  f1o0  5618  en0  6964