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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 5669 | . . 3 | |
2 | fveq2 5421 | . . . . . . . 8 | |
3 | 2 | eqeq1d 2148 | . . . . . . 7 |
4 | eqeq1 2146 | . . . . . . 7 | |
5 | 3, 4 | imbi12d 233 | . . . . . 6 |
6 | fveq2 5421 | . . . . . . . 8 | |
7 | 6 | eqeq2d 2151 | . . . . . . 7 |
8 | eqeq2 2149 | . . . . . . 7 | |
9 | 7, 8 | imbi12d 233 | . . . . . 6 |
10 | 5, 9 | rspc2v 2802 | . . . . 5 |
11 | 10 | com12 30 | . . . 4 |
12 | 11 | adantl 275 | . . 3 |
13 | 1, 12 | sylbi 120 | . 2 |
14 | 13 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 wf 5119 wf1 5120 cfv 5123 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fv 5131 |
This theorem is referenced by: f1fveq 5673 f1ocnvfvrneq 5683 f1o2ndf1 6125 fidceq 6763 difinfsnlem 6984 difinfsn 6985 iseqf1olemab 10262 iseqf1olemnanb 10263 pwle2 13193 |
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