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Mirrors > Home > NFE Home > Th. List > fsn | Unicode version |
Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.) |
Ref | Expression |
---|---|
fsn.1 | |
fsn.2 |
Ref | Expression |
---|---|
fsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelf 5235 | . . . . . . 7 | |
2 | elsn 3748 | . . . . . . . 8 | |
3 | elsn 3748 | . . . . . . . 8 | |
4 | 2, 3 | anbi12i 678 | . . . . . . 7 |
5 | 1, 4 | sylib 188 | . . . . . 6 |
6 | 5 | ex 423 | . . . . 5 |
7 | fsn.1 | . . . . . . . . 9 | |
8 | 7 | snid 3760 | . . . . . . . 8 |
9 | feu 5242 | . . . . . . . 8 | |
10 | 8, 9 | mpan2 652 | . . . . . . 7 |
11 | 3 | anbi1i 676 | . . . . . . . . . 10 |
12 | opeq2 4579 | . . . . . . . . . . . . 13 | |
13 | 12 | eleq1d 2419 | . . . . . . . . . . . 12 |
14 | 13 | pm5.32i 618 | . . . . . . . . . . 11 |
15 | ancom 437 | . . . . . . . . . . 11 | |
16 | 14, 15 | bitr4i 243 | . . . . . . . . . 10 |
17 | 11, 16 | bitr2i 241 | . . . . . . . . 9 |
18 | 17 | eubii 2213 | . . . . . . . 8 |
19 | fsn.2 | . . . . . . . . . . 11 | |
20 | 19 | eueq1 3009 | . . . . . . . . . 10 |
21 | 20 | biantru 491 | . . . . . . . . 9 |
22 | euanv 2265 | . . . . . . . . 9 | |
23 | 21, 22 | bitr4i 243 | . . . . . . . 8 |
24 | df-reu 2621 | . . . . . . . 8 | |
25 | 18, 23, 24 | 3bitr4i 268 | . . . . . . 7 |
26 | 10, 25 | sylibr 203 | . . . . . 6 |
27 | opeq12 4580 | . . . . . . 7 | |
28 | 27 | eleq1d 2419 | . . . . . 6 |
29 | 26, 28 | syl5ibrcom 213 | . . . . 5 |
30 | 6, 29 | impbid 183 | . . . 4 |
31 | vex 2862 | . . . . . . 7 | |
32 | vex 2862 | . . . . . . 7 | |
33 | 31, 32 | opex 4588 | . . . . . 6 |
34 | 33 | elsnc 3756 | . . . . 5 |
35 | opth 4602 | . . . . 5 | |
36 | 34, 35 | bitr2i 241 | . . . 4 |
37 | 30, 36 | syl6bb 252 | . . 3 |
38 | 37 | eqrelrdv 4852 | . 2 |
39 | 7, 19 | f1osn 5322 | . . . 4 |
40 | f1oeq1 5281 | . . . 4 | |
41 | 39, 40 | mpbiri 224 | . . 3 |
42 | f1of 5287 | . . 3 | |
43 | 41, 42 | syl 15 | . 2 |
44 | 38, 43 | impbii 180 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 wceq 1642 wcel 1710 weu 2204 wreu 2616 cvv 2859 csn 3737 cop 4561 wf 4777 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-id 4767 df-xp 4784 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: fsng 5433 fsn2 5434 xpsn 5435 mapsn 6026 |
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