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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 4811 | . . . 4 | |
3 | 1, 2 | feq12d 5337 | . . 3 |
4 | ordeq 4357 | . . . 4 | |
5 | 2, 4 | syl 14 | . . 3 |
6 | fveq1 5495 | . . . . . . 7 | |
7 | fveq1 5495 | . . . . . . 7 | |
8 | 6, 7 | eleq12d 2241 | . . . . . 6 |
9 | 8 | imbi2d 229 | . . . . 5 |
10 | 9 | 2ralbidv 2494 | . . . 4 |
11 | 2 | raleqdv 2671 | . . . . 5 |
12 | 11 | ralbidv 2470 | . . . 4 |
13 | 2 | raleqdv 2671 | . . . 4 |
14 | 10, 12, 13 | 3bitrd 213 | . . 3 |
15 | 3, 5, 14 | 3anbi123d 1307 | . 2 |
16 | df-smo 6265 | . 2 | |
17 | df-smo 6265 | . 2 | |
18 | 15, 16, 17 | 3bitr4g 222 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 973 wceq 1348 wcel 2141 wral 2448 word 4347 con0 4348 cdm 4611 wf 5194 cfv 5198 wsmo 6264 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-tr 4088 df-iord 4351 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-smo 6265 |
This theorem is referenced by: smores3 6272 smo0 6277 |
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