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Mirrors > Home > MPE Home > Th. List > mapsnf1o2 | Structured version Visualization version Unicode version |
Description: Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
mapsncnv.s | |
mapsncnv.b | |
mapsncnv.x | |
mapsncnv.f |
Ref | Expression |
---|---|
mapsnf1o2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6201 | . . 3 | |
2 | mapsncnv.f | . . 3 | |
3 | 1, 2 | fnmpti 6022 | . 2 |
4 | mapsncnv.s | . . . . 5 | |
5 | snex 4908 | . . . . 5 | |
6 | 4, 5 | eqeltri 2697 | . . . 4 |
7 | snex 4908 | . . . 4 | |
8 | 6, 7 | xpex 6962 | . . 3 |
9 | mapsncnv.b | . . . 4 | |
10 | mapsncnv.x | . . . 4 | |
11 | 4, 9, 10, 2 | mapsncnv 7904 | . . 3 |
12 | 8, 11 | fnmpti 6022 | . 2 |
13 | dff1o4 6145 | . 2 | |
14 | 3, 12, 13 | mpbir2an 955 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cvv 3200 csn 4177 cmpt 4729 cxp 5112 ccnv 5113 wfn 5883 wf1o 5887 cfv 5888 (class class class)co 6650 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 |
This theorem is referenced by: mapsnf1o3 7906 coe1sfi 19583 coe1mul2lem2 19638 ply1coe 19666 evl1var 19700 pf1mpf 19716 pf1ind 19719 deg1ldg 23852 deg1leb 23855 |
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