Theorem fo00 3706

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 3663	. . . . . 6 |- (F:(/)-onto->A -> F:(/)-->A)
2		ffn 3619	. . . . . 6 |- (F:(/)-->A -> F Fn (/))
3		fn0 3597	. . . . . . 7 |- (F Fn (/) <-> F = (/))
4		f10 3704	. . . . . . . 8 |- (/):(/)-1-1->A
5		f1eq1 3651	. . . . . . . 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 3192	. . . 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 3686	. . 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 3705	. 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 954  (/)c0 2276   Fn wfn 3172  -->wf 3173  -1-1->wf1 3174  -onto->wfo 3175  -1-1-onto->wf1o 3176

This theorem is referenced by:  fodomfi 4546

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192

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