Theorem fo00 3721

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		fof 3678	. . . . . 6 |- (F:(/)-onto->A -> F:(/)-->A)
2		ffn 3633	. . . . . 6 |- (F:(/)-->A -> F Fn (/))
3		fn0 3611	. . . . . . 7 |- (F Fn (/) <-> F = (/))
4		f10 3719	. . . . . . . 8 |- (/):(/)-1-1->A
5		f1eq1 3666	. . . . . . . 8 |- (F = (/) -> (F:(/)-1-1->A <-> (/):(/)-1-1->A))
6	4, 5	mpbiri 194	. . . . . . 7 |- (F = (/) -> F:(/)-1-1->A)
7	3, 6	sylbi 199	. . . . . 6 |- (F Fn (/) -> F:(/)-1-1->A)
8	1, 2, 7	3syl 20	. . . . 5 |- (F:(/)-onto->A -> F:(/)-1-1->A)
9	8	ancri 297	. . . 4 |- (F:(/)-onto->A -> (F:(/)-1-1->A /\ F:(/)-onto->A))
10		df-f1o 3203	. . . 4 |- (F:(/)-1-1-onto->A <-> (F:(/)-1-1->A /\ F:(/)-onto->A))
11	9, 10	sylibr 200	. . 3 |- (F:(/)-onto->A -> F:(/)-1-1-onto->A)
12		f1ofo 3701	. . 3 |- (F:(/)-1-1-onto->A -> F:(/)-onto->A)
13	11, 12	impbi 157	. 2 |- (F:(/)-onto->A <-> F:(/)-1-1-onto->A)
14		f1o00 3720	. 2 |- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))
15	13, 14	bitr 173	1 |- (F:(/)-onto->A <-> (F = (/) /\ A = (/)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958  (/)c0 2283   Fn wfn 3183  -->wf 3184  -1-1->wf1 3185  -onto->wfo 3186  -1-1-onto->wf1o 3187

This theorem is referenced by:  fodomfi 4575  fodomfiOLD 4576

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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