Theorem f1o00 3720

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 3702	. 2 |- (F:(/)-1-1-onto->A <-> (F Fn (/) /\ `'F Fn A))
2		fn0 3611	. . . . . 6 |- (F Fn (/) <-> F = (/))
3	2	biimp 151	. . . . 5 |- (F Fn (/) -> F = (/))
4	3	adantr 391	. . . 4 |- ((F Fn (/) /\ `'F Fn A) -> F = (/))
5		cnveq 3298	. . . . . . . . . 10 |- (F = (/) -> `'F = `'(/))
6		cnv0 3452	. . . . . . . . . 10 |- `'(/) = (/)
7	5, 6	syl6eq 1526	. . . . . . . . 9 |- (F = (/) -> `'F = (/))
8	2, 7	sylbi 199	. . . . . . . 8 |- (F Fn (/) -> `'F = (/))
9		fneq1 3588	. . . . . . . 8 |- (`'F = (/) -> (`'F Fn A <-> (/) Fn A))
10	8, 9	syl 10	. . . . . . 7 |- (F Fn (/) -> (`'F Fn A <-> (/) Fn A))
11	10	biimpa 418	. . . . . 6 |- ((F Fn (/) /\ `'F Fn A) -> (/) Fn A)
12		fndm 3593	. . . . . 6 |- ((/) Fn A -> dom (/) = A)
13	11, 12	syl 10	. . . . 5 |- ((F Fn (/) /\ `'F Fn A) -> dom (/) = A)
14		dm0 3329	. . . . 5 |- dom (/) = (/)
15	13, 14	syl5reqr 1525	. . . 4 |- ((F Fn (/) /\ `'F Fn A) -> A = (/))
16	4, 15	jca 288	. . 3 |- ((F Fn (/) /\ `'F Fn A) -> (F = (/) /\ A = (/)))
17	2	biimpr 152	. . . . 5 |- (F = (/) -> F Fn (/))
18	17	adantr 391	. . . 4 |- ((F = (/) /\ A = (/)) -> F Fn (/))
19		eqid 1478	. . . . . 6 |- (/) = (/)
20		fn0 3611	. . . . . 6 |- ((/) Fn (/) <-> (/) = (/))
21	19, 20	mpbir 190	. . . . 5 |- (/) Fn (/)
22	7, 9	syl 10	. . . . . 6 |- (F = (/) -> (`'F Fn A <-> (/) Fn A))
23		fneq2 3589	. . . . . 6 |- (A = (/) -> ((/) Fn A <-> (/) Fn (/)))
24	22, 23	sylan9bb 542	. . . . 5 |- ((F = (/) /\ A = (/)) -> (`'F Fn A <-> (/) Fn (/)))
25	21, 24	mpbiri 194	. . . 4 |- ((F = (/) /\ A = (/)) -> `'F Fn A)
26	18, 25	jca 288	. . 3 |- ((F = (/) /\ A = (/)) -> (F Fn (/) /\ `'F Fn A))
27	16, 26	impbi 157	. 2 |- ((F Fn (/) /\ `'F Fn A) <-> (F = (/) /\ A = (/)))
28	1, 27	bitr 173	1 |- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958  (/)c0 2283  `'ccnv 3175  dom cdm 3176   Fn wfn 3183  -1-1-onto->wf1o 3187

This theorem is referenced by:  fo00 3721  f1o0 3722  en0 4429

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203

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