Theorem fo00 4669

Description: Onto mapping of the empty set.

Assertion

Ref	Expression
fo00	|- (F:(/)-onto->A <-> (F = (/) /\ A = (/)))

Proof of Theorem fo00

Step	Hyp	Ref	Expression
1		fofn 4619	. . . . . 6 |- (F:(/)-onto->A -> F Fn (/))
2		fn0 4532	. . . . . . 7 |- (F Fn (/) <-> F = (/))
3		f10 4667	. . . . . . . 8 |- (/):(/)-1-1->A
4		f1eq1 4605	. . . . . . . 8 |- (F = (/) -> (F:(/)-1-1->A <-> (/):(/)-1-1->A))
5	3, 4	mpbiri 211	. . . . . . 7 |- (F = (/) -> F:(/)-1-1->A)
6	2, 5	sylbi 216	. . . . . 6 |- (F Fn (/) -> F:(/)-1-1->A)
7	1, 6	syl 12	. . . . 5 |- (F:(/)-onto->A -> F:(/)-1-1->A)
8	7	ancri 321	. . . 4 |- (F:(/)-onto->A -> (F:(/)-1-1->A /\ F:(/)-onto->A))
9		df-f1o 4013	. . . 4 |- (F:(/)-1-1-onto->A <-> (F:(/)-1-1->A /\ F:(/)-onto->A))
10	8, 9	sylibr 217	. . 3 |- (F:(/)-onto->A -> F:(/)-1-1-onto->A)
11		f1ofo 4643	. . 3 |- (F:(/)-1-1-onto->A -> F:(/)-onto->A)
12	10, 11	impbii 174	. 2 |- (F:(/)-onto->A <-> F:(/)-1-1-onto->A)
13		f1o00 4668	. 2 |- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))
14	12, 13	bitri 190	1 |- (F:(/)-onto->A <-> (F = (/) /\ A = (/)))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298  (/)c0 2875   Fn wfn 3993  -1-1->wf1 3995  -onto->wfo 3996  -1-1-onto->wf1o 3997

This theorem is referenced by:  fodomfi 5656

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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