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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff13 5790 |
. . 3
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2 | fveq2 5534 |
. . . . . . . 8
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3 | 2 | eqeq1d 2198 |
. . . . . . 7
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4 | eqeq1 2196 |
. . . . . . 7
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5 | 3, 4 | imbi12d 234 |
. . . . . 6
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6 | fveq2 5534 |
. . . . . . . 8
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7 | 6 | eqeq2d 2201 |
. . . . . . 7
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8 | eqeq2 2199 |
. . . . . . 7
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9 | 7, 8 | imbi12d 234 |
. . . . . 6
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10 | 5, 9 | rspc2v 2869 |
. . . . 5
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11 | 10 | com12 30 |
. . . 4
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12 | 11 | adantl 277 |
. . 3
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13 | 1, 12 | sylbi 121 |
. 2
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14 | 13 | imp 124 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fv 5243 |
This theorem is referenced by: f1fveq 5794 f1ocnvfvrneq 5804 f1o2ndf1 6254 fidceq 6898 difinfsnlem 7129 difinfsn 7130 iseqf1olemab 10522 iseqf1olemnanb 10523 f1ghm0to0 13228 1dom1el 15221 pwle2 15227 |
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