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Mirrors > Home > MPE Home > Th. List > f1veqaeq | Structured version Visualization version 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 6512 | . . 3 | |
2 | fveq2 6191 | . . . . . . . 8 | |
3 | 2 | eqeq1d 2624 | . . . . . . 7 |
4 | eqeq1 2626 | . . . . . . 7 | |
5 | 3, 4 | imbi12d 334 | . . . . . 6 |
6 | fveq2 6191 | . . . . . . . 8 | |
7 | 6 | eqeq2d 2632 | . . . . . . 7 |
8 | eqeq2 2633 | . . . . . . 7 | |
9 | 7, 8 | imbi12d 334 | . . . . . 6 |
10 | 5, 9 | rspc2v 3322 | . . . . 5 |
11 | 10 | com12 32 | . . . 4 |
12 | 11 | adantl 482 | . . 3 |
13 | 1, 12 | sylbi 207 | . 2 |
14 | 13 | imp 445 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 wf 5884 wf1 5885 cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fv 5896 |
This theorem is referenced by: f1cofveqaeq 6515 f1cofveqaeqALT 6516 2f1fvneq 6517 f1fveq 6519 f1prex 6539 f1ocnvfvrneq 6541 f1o2ndf1 7285 symgfvne 17808 f1rhm0to0 18740 mat2pmatf1 20534 f1otrg 25751 uspgr2wlkeq 26542 pthdivtx 26625 spthdep 26630 spthonepeq 26648 usgr2trlncl 26656 poimirlem1 33410 poimirlem9 33418 poimirlem22 33431 mblfinlem2 33447 |
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