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Mirrors > Home > ILE Home > Th. List > fun11 | Unicode version |
Description: Two ways of stating that is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.) |
Ref | Expression |
---|---|
fun11 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi2 388 | . . . . . . . 8 | |
2 | 1 | imbi2i 226 | . . . . . . 7 |
3 | pm4.76 604 | . . . . . . 7 | |
4 | bi2.04 248 | . . . . . . . 8 | |
5 | bi2.04 248 | . . . . . . . 8 | |
6 | 4, 5 | anbi12i 460 | . . . . . . 7 |
7 | 2, 3, 6 | 3bitr2i 208 | . . . . . 6 |
8 | 7 | 2albii 1469 | . . . . 5 |
9 | 19.26-2 1480 | . . . . 5 | |
10 | alcom 1476 | . . . . . . 7 | |
11 | nfv 1526 | . . . . . . . . 9 | |
12 | breq1 4001 | . . . . . . . . . . 11 | |
13 | 12 | anbi1d 465 | . . . . . . . . . 10 |
14 | 13 | imbi1d 231 | . . . . . . . . 9 |
15 | 11, 14 | equsal 1725 | . . . . . . . 8 |
16 | 15 | albii 1468 | . . . . . . 7 |
17 | 10, 16 | bitri 184 | . . . . . 6 |
18 | nfv 1526 | . . . . . . . 8 | |
19 | breq2 4002 | . . . . . . . . . 10 | |
20 | 19 | anbi1d 465 | . . . . . . . . 9 |
21 | 20 | imbi1d 231 | . . . . . . . 8 |
22 | 18, 21 | equsal 1725 | . . . . . . 7 |
23 | 22 | albii 1468 | . . . . . 6 |
24 | 17, 23 | anbi12i 460 | . . . . 5 |
25 | 8, 9, 24 | 3bitri 206 | . . . 4 |
26 | 25 | 2albii 1469 | . . 3 |
27 | 19.26-2 1480 | . . 3 | |
28 | 26, 27 | bitr2i 185 | . 2 |
29 | fun2cnv 5272 | . . . 4 | |
30 | breq2 4002 | . . . . . 6 | |
31 | 30 | mo4 2085 | . . . . 5 |
32 | 31 | albii 1468 | . . . 4 |
33 | alcom 1476 | . . . . 5 | |
34 | 33 | albii 1468 | . . . 4 |
35 | 29, 32, 34 | 3bitri 206 | . . 3 |
36 | funcnv2 5268 | . . . 4 | |
37 | breq1 4001 | . . . . . 6 | |
38 | 37 | mo4 2085 | . . . . 5 |
39 | 38 | albii 1468 | . . . 4 |
40 | alcom 1476 | . . . . . 6 | |
41 | 40 | albii 1468 | . . . . 5 |
42 | alcom 1476 | . . . . 5 | |
43 | 41, 42 | bitri 184 | . . . 4 |
44 | 36, 39, 43 | 3bitri 206 | . . 3 |
45 | 35, 44 | anbi12i 460 | . 2 |
46 | alrot4 1484 | . 2 | |
47 | 28, 45, 46 | 3bitr4i 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wal 1351 wmo 2025 class class class wbr 3998 ccnv 4619 wfun 5202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-fun 5210 |
This theorem is referenced by: (None) |
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