Theorem f1o00 2823

Description: One-to-one onto mapping of the empty set.

Assertion

Ref	Expression
f1o00	|- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))

Proof of Theorem f1o00

Step	Hyp	Ref	Expression
1		f1o4 2807	. 2 |- (F:(/)-1-1-onto->A <-> (F Fn (/) /\ `'F Fn A))
2		fn0 2739	. . . . . 6 |- (F Fn (/) <-> F = (/))
3	2	biimp 133	. . . . 5 |- (F Fn (/) -> F = (/))
4	3	adantr 306	. . . 4 |- ((F Fn (/) /\ `'F Fn A) -> F = (/))
5		cnveq 2513	. . . . . . . . . 10 |- (F = (/) -> `'F = `'(/))
6		cnv0 2633	. . . . . . . . . 10 |- `'(/) = (/)
7	5, 6	syl6eq 1140	. . . . . . . . 9 |- (F = (/) -> `'F = (/))
8	2, 7	sylbi 174	. . . . . . . 8 |- (F Fn (/) -> `'F = (/))
9		fneq1 2718	. . . . . . . 8 |- (`'F = (/) -> (`'F Fn A <-> (/) Fn A))
10	8, 9	syl 12	. . . . . . 7 |- (F Fn (/) -> (`'F Fn A <-> (/) Fn A))
11	10	biimpa 324	. . . . . 6 |- ((F Fn (/) /\ `'F Fn A) -> (/) Fn A)
12		fndm 2723	. . . . . 6 |- ((/) Fn A -> dom (/) = A)
13	11, 12	syl 12	. . . . 5 |- ((F Fn (/) /\ `'F Fn A) -> dom (/) = A)
14		dm0 2542	. . . . 5 |- dom (/) = (/)
15	13, 14	syl5reqr 1139	. . . 4 |- ((F Fn (/) /\ `'F Fn A) -> A = (/))
16	4, 15	jca 236	. . 3 |- ((F Fn (/) /\ `'F Fn A) -> (F = (/) /\ A = (/)))
17	2	biimpr 134	. . . . 5 |- (F = (/) -> F Fn (/))
18	17	adantr 306	. . . 4 |- ((F = (/) /\ A = (/)) -> F Fn (/))
19		cleqid 1102	. . . . . 6 |- (/) = (/)
20		fn0 2739	. . . . . 6 |- ((/) Fn (/) <-> (/) = (/))
21	19, 20	mpbir 165	. . . . 5 |- (/) Fn (/)
22	7, 9	syl 12	. . . . . 6 |- (F = (/) -> (`'F Fn A <-> (/) Fn A))
23		fneq2 2719	. . . . . 6 |- (A = (/) -> ((/) Fn A <-> (/) Fn (/)))
24	22, 23	sylan9bb 418	. . . . 5 |- ((F = (/) /\ A = (/)) -> (`'F Fn A <-> (/) Fn (/)))
25	21, 24	mpbiri 169	. . . 4 |- ((F = (/) /\ A = (/)) -> `'F Fn A)
26	18, 25	jca 236	. . 3 |- ((F = (/) /\ A = (/)) -> (F Fn (/) /\ `'F Fn A))
27	16, 26	impbi 139	. 2 |- ((F Fn (/) /\ `'F Fn A) <-> (F = (/) /\ A = (/)))
28	1, 27	bitr 151	1 |- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091  (/)c0 1707  `'ccnv 2409  dom cdm 2410   Fn wfn 2417  -1-1-onto->wf1o 2421

This theorem is referenced by:  f1o0 2824  en0 3328

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

metamath.org