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Mirrors > Home > ILE Home > Th. List > f1veqaeq | Unicode version |
Description: If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.) |
Ref | Expression |
---|---|
f1veqaeq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff13 5407 | . . 3 | |
2 | fveq2 5178 | . . . . . . . 8 | |
3 | 2 | eqeq1d 2048 | . . . . . . 7 |
4 | eqeq1 2046 | . . . . . . 7 | |
5 | 3, 4 | imbi12d 223 | . . . . . 6 |
6 | fveq2 5178 | . . . . . . . 8 | |
7 | 6 | eqeq2d 2051 | . . . . . . 7 |
8 | eqeq2 2049 | . . . . . . 7 | |
9 | 7, 8 | imbi12d 223 | . . . . . 6 |
10 | 5, 9 | rspc2v 2662 | . . . . 5 |
11 | 10 | com12 27 | . . . 4 |
12 | 11 | adantl 262 | . . 3 |
13 | 1, 12 | sylbi 114 | . 2 |
14 | 13 | imp 115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 wral 2306 wf 4898 wf1 4899 cfv 4902 |
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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fv 4910 |
This theorem is referenced by: f1fveq 5411 f1ocnvfvrneq 5422 f1o2ndf1 5849 fidceq 6330 |
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