Theorem f1o00 3709

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 3691	. 2 |- (F:(/)-1-1-onto->A <-> (F Fn (/) /\ `'F Fn A))
2		fn0 3601	. . . . . 6 |- (F Fn (/) <-> F = (/))
3	2	biimp 151	. . . . 5 |- (F Fn (/) -> F = (/))
4	3	adantr 389	. . . 4 |- ((F Fn (/) /\ `'F Fn A) -> F = (/))
5		cnveq 3288	. . . . . . . . . 10 |- (F = (/) -> `'F = `'(/))
6		cnv0 3442	. . . . . . . . . 10 |- `'(/) = (/)
7	5, 6	syl6eq 1521	. . . . . . . . 9 |- (F = (/) -> `'F = (/))
8	2, 7	sylbi 199	. . . . . . . 8 |- (F Fn (/) -> `'F = (/))
9		fneq1 3578	. . . . . . . 8 |- (`'F = (/) -> (`'F Fn A <-> (/) Fn A))
10	8, 9	syl 10	. . . . . . 7 |- (F Fn (/) -> (`'F Fn A <-> (/) Fn A))
11	10	biimpa 416	. . . . . 6 |- ((F Fn (/) /\ `'F Fn A) -> (/) Fn A)
12		fndm 3583	. . . . . 6 |- ((/) Fn A -> dom (/) = A)
13	11, 12	syl 10	. . . . 5 |- ((F Fn (/) /\ `'F Fn A) -> dom (/) = A)
14		dm0 3319	. . . . 5 |- dom (/) = (/)
15	13, 14	syl5reqr 1520	. . . 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 389	. . . 4 |- ((F = (/) /\ A = (/)) -> F Fn (/))
19		eqid 1474	. . . . . 6 |- (/) = (/)
20		fn0 3601	. . . . . 6 |- ((/) Fn (/) <-> (/) = (/))
21	19, 20	mpbir 190	. . . . 5 |- (/) Fn (/)
22	7, 9	syl 10	. . . . . 6 |- (F = (/) -> (`'F Fn A <-> (/) Fn A))
23		fneq2 3579	. . . . . 6 |- (A = (/) -> ((/) Fn A <-> (/) Fn (/)))
24	22, 23	sylan9bb 539	. . . . 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 955  (/)c0 2277  `'ccnv 3165  dom cdm 3166   Fn wfn 3173  -1-1-onto->wf1o 3177

This theorem is referenced by:  fo00 3710  f1o0 3711  en0 4413

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193

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