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Theorem f1o00 5317

Description: One-to-one onto mapping of the empty set. (Contributed by set.mm contributors, 15-Apr-1998.)

**Assertion**
Ref	Expression
f1o00	$/\$

**Proof of Theorem f1o00**
Step	Hyp	Ref	Expression
1		dff1o4 5294	. 2 $/\$
2		fn0 5202	. . . . . 6
3	2	biimpi 186	. . . . 5
4	3	adantr 451	. . . 4 $/\$
5		dm0 4918	. . . . 5
6		cnveq 4886	. . . . . . . . . 10
7		cnv0 5031	. . . . . . . . . 10
8	6, 7	syl6eq 2401	. . . . . . . . 9
9	2, 8	sylbi 187	. . . . . . . 8
10	9	fneq1d 5175	. . . . . . 7
11	10	biimpa 470	. . . . . 6 $/\$
12		fndm 5182	. . . . . 6
13	11, 12	syl 15	. . . . 5 $/\$
14	5, 13	syl5reqr 2400	. . . 4 $/\$
15	4, 14	jca 518	. . 3 $/\$ $/\$
16	2	biimpri 197	. . . . 5
17	16	adantr 451	. . . 4 $/\$
18		eqid 2353	. . . . . 6
19		fn0 5202	. . . . . 6
20	18, 19	mpbir 200	. . . . 5
21	8	fneq1d 5175	. . . . . 6
22		fneq2 5174	. . . . . 6
23	21, 22	sylan9bb 680	. . . . 5 $/\$
24	20, 23	mpbiri 224	. . . 4 $/\$
25	17, 24	jca 518	. . 3 $/\$ $/\$
26	15, 25	impbii 180	. 2 $/\$ $/\$
27	1, 26	bitri 240	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wb 176 $/\$ wa 358

wceq 1642

c0 3550

ccnv 4771

cdm 4772

wfn 4776

wf1o 4780

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794

This theorem is referenced by: fo00 5318 en0 6042