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Mirrors > Home > ILE Home > Th. List > smoeq | Unicode version |
Description: Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Ref | Expression |
---|---|
smoeq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . 4 | |
2 | dmeq 4535 | . . . 4 | |
3 | 1, 2 | feq12d 5036 | . . 3 |
4 | ordeq 4109 | . . . 4 | |
5 | 2, 4 | syl 14 | . . 3 |
6 | fveq1 5177 | . . . . . . 7 | |
7 | fveq1 5177 | . . . . . . 7 | |
8 | 6, 7 | eleq12d 2108 | . . . . . 6 |
9 | 8 | imbi2d 219 | . . . . 5 |
10 | 9 | 2ralbidv 2348 | . . . 4 |
11 | 2 | raleqdv 2511 | . . . . 5 |
12 | 11 | ralbidv 2326 | . . . 4 |
13 | 2 | raleqdv 2511 | . . . 4 |
14 | 10, 12, 13 | 3bitrd 203 | . . 3 |
15 | 3, 5, 14 | 3anbi123d 1207 | . 2 |
16 | df-smo 5901 | . 2 | |
17 | df-smo 5901 | . 2 | |
18 | 15, 16, 17 | 3bitr4g 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 w3a 885 wceq 1243 wcel 1393 wral 2306 word 4099 con0 4100 cdm 4345 wf 4898 cfv 4902 wsmo 5900 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-tr 3855 df-iord 4103 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-smo 5901 |
This theorem is referenced by: smores3 5908 smo0 5913 |
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