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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 4709 | . . . 4 | |
3 | 1, 2 | feq12d 5232 | . . 3 |
4 | ordeq 4264 | . . . 4 | |
5 | 2, 4 | syl 14 | . . 3 |
6 | fveq1 5388 | . . . . . . 7 | |
7 | fveq1 5388 | . . . . . . 7 | |
8 | 6, 7 | eleq12d 2188 | . . . . . 6 |
9 | 8 | imbi2d 229 | . . . . 5 |
10 | 9 | 2ralbidv 2436 | . . . 4 |
11 | 2 | raleqdv 2609 | . . . . 5 |
12 | 11 | ralbidv 2414 | . . . 4 |
13 | 2 | raleqdv 2609 | . . . 4 |
14 | 10, 12, 13 | 3bitrd 213 | . . 3 |
15 | 3, 5, 14 | 3anbi123d 1275 | . 2 |
16 | df-smo 6151 | . 2 | |
17 | df-smo 6151 | . 2 | |
18 | 15, 16, 17 | 3bitr4g 222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 947 wceq 1316 wcel 1465 wral 2393 word 4254 con0 4255 cdm 4509 wf 5089 cfv 5093 wsmo 6150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-tr 3997 df-iord 4258 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-smo 6151 |
This theorem is referenced by: smores3 6158 smo0 6163 |
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