Theorem f1o00 6171

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 6145	. 2 |- ( F : (/) -1-1-onto-> A <-> ( F Fn (/) /\ `' F Fn A ) )
2		fn0 6011	. . . . . 6 |- ( F Fn (/) <-> F = (/) )
3	2	biimpi 206	. . . . 5 |- ( F Fn (/) -> F = (/) )
4	3	adantr 481	. . . 4 |- ( ( F Fn (/) /\ `' F Fn A ) -> F = (/) )
5		dm0 5339	. . . . 5 |- dom (/) = (/)
6		cnveq 5296	. . . . . . . . . 10 |- ( F = (/) -> `' F = `' (/) )
7		cnv0 5535	. . . . . . . . . 10 |- `' (/) = (/)
8	6, 7	syl6eq 2672	. . . . . . . . 9 |- ( F = (/) -> `' F = (/) )
9	2, 8	sylbi 207	. . . . . . . 8 |- ( F Fn (/) -> `' F = (/) )
10	9	fneq1d 5981	. . . . . . 7 |- ( F Fn (/) -> ( `' F Fn A <-> (/) Fn A ) )
11	10	biimpa 501	. . . . . 6 |- ( ( F Fn (/) /\ `' F Fn A ) -> (/) Fn A )
12		fndm 5990	. . . . . 6 |- ( (/) Fn A -> dom (/) = A )
13	11, 12	syl 17	. . . . 5 |- ( ( F Fn (/) /\ `' F Fn A ) -> dom (/) = A )
14	5, 13	syl5reqr 2671	. . . 4 |- ( ( F Fn (/) /\ `' F Fn A ) -> A = (/) )
15	4, 14	jca 554	. . 3 |- ( ( F Fn (/) /\ `' F Fn A ) -> ( F = (/) /\ A = (/) ) )
16	2	biimpri 218	. . . . 5 |- ( F = (/) -> F Fn (/) )
17	16	adantr 481	. . . 4 |- ( ( F = (/) /\ A = (/) ) -> F Fn (/) )
18		eqid 2622	. . . . . 6 |- (/) = (/)
19		fn0 6011	. . . . . 6 |- ( (/) Fn (/) <-> (/) = (/) )
20	18, 19	mpbir 221	. . . . 5 |- (/) Fn (/)
21	8	fneq1d 5981	. . . . . 6 |- ( F = (/) -> ( `' F Fn A <-> (/) Fn A ) )
22		fneq2 5980	. . . . . 6 |- ( A = (/) -> ( (/) Fn A <-> (/) Fn (/) ) )
23	21, 22	sylan9bb 736	. . . . 5 |- ( ( F = (/) /\ A = (/) ) -> ( `' F Fn A <-> (/) Fn (/) ) )
24	20, 23	mpbiri 248	. . . 4 |- ( ( F = (/) /\ A = (/) ) -> `' F Fn A )
25	17, 24	jca 554	. . 3 |- ( ( F = (/) /\ A = (/) ) -> ( F Fn (/) /\ `' F Fn A ) )
26	15, 25	impbii 199	. 2 |- ( ( F Fn (/) /\ `' F Fn A ) <-> ( F = (/) /\ A = (/) ) )
27	1, 26	bitri 264	1 |- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   (/)c0 3915   `'ccnv 5113   dom cdm 5114   Fn wfn 5883   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  fo00  6172  f1o0  6173  en0  8019  symgbas0  17814  derang0  31151  poimirlem28  33437