Theorem f1o00 5467

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 5440	. 2 |- ( F : (/) -1-1-onto-> A <-> ( F Fn (/) /\ `' F Fn A ) )
2		fn0 5307	. . . . . 6 |- ( F Fn (/) <-> F = (/) )
3	2	biimpi 119	. . . . 5 |- ( F Fn (/) -> F = (/) )
4	3	adantr 274	. . . 4 |- ( ( F Fn (/) /\ `' F Fn A ) -> F = (/) )
5		cnveq 4778	. . . . . . . . . 10 |- ( F = (/) -> `' F = `' (/) )
6		cnv0 5007	. . . . . . . . . 10 |- `' (/) = (/)
7	5, 6	eqtrdi 2215	. . . . . . . . 9 |- ( F = (/) -> `' F = (/) )
8	2, 7	sylbi 120	. . . . . . . 8 |- ( F Fn (/) -> `' F = (/) )
9	8	fneq1d 5278	. . . . . . 7 |- ( F Fn (/) -> ( `' F Fn A <-> (/) Fn A ) )
10	9	biimpa 294	. . . . . 6 |- ( ( F Fn (/) /\ `' F Fn A ) -> (/) Fn A )
11		fndm 5287	. . . . . 6 |- ( (/) Fn A -> dom (/) = A )
12	10, 11	syl 14	. . . . 5 |- ( ( F Fn (/) /\ `' F Fn A ) -> dom (/) = A )
13		dm0 4818	. . . . 5 |- dom (/) = (/)
14	12, 13	eqtr3di 2214	. . . 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 2165	. . . . . 6 |- (/) = (/)
19		fn0 5307	. . . . . 6 |- ( (/) Fn (/) <-> (/) = (/) )
20	18, 19	mpbir 145	. . . . 5 |- (/) Fn (/)
21	7	fneq1d 5278	. . . . . 6 |- ( F = (/) -> ( `' F Fn A <-> (/) Fn A ) )
22		fneq2 5277	. . . . . 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 1343   (/)c0 3409   `'ccnv 4603   dom cdm 4604   Fn wfn 5183   -1-1-onto->wf1o 5187

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by:  fo00  5468  f1o0  5469  en0  6761