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Mirrors > Home > MPE Home > Th. List > pmtrprfv | Structured version Visualization version Unicode version |
Description: In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
Ref | Expression |
---|---|
pmtrfval.t | pmTrsp |
Ref | Expression |
---|---|
pmtrprfv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . 3 | |
2 | simpr1 1067 | . . . 4 | |
3 | simpr2 1068 | . . . 4 | |
4 | prssi 4353 | . . . 4 | |
5 | 2, 3, 4 | syl2anc 693 | . . 3 |
6 | pr2nelem 8827 | . . . 4 | |
7 | 6 | adantl 482 | . . 3 |
8 | pmtrfval.t | . . . 4 pmTrsp | |
9 | 8 | pmtrfv 17872 | . . 3 |
10 | 1, 5, 7, 2, 9 | syl31anc 1329 | . 2 |
11 | prid1g 4295 | . . . . 5 | |
12 | 2, 11 | syl 17 | . . . 4 |
13 | 12 | iftrued 4094 | . . 3 |
14 | difprsnss 4329 | . . . . . . 7 | |
15 | 14 | a1i 11 | . . . . . 6 |
16 | prid2g 4296 | . . . . . . . . 9 | |
17 | 3, 16 | syl 17 | . . . . . . . 8 |
18 | simpr3 1069 | . . . . . . . . 9 | |
19 | 18 | necomd 2849 | . . . . . . . 8 |
20 | eldifsn 4317 | . . . . . . . 8 | |
21 | 17, 19, 20 | sylanbrc 698 | . . . . . . 7 |
22 | 21 | snssd 4340 | . . . . . 6 |
23 | 15, 22 | eqssd 3620 | . . . . 5 |
24 | 23 | unieqd 4446 | . . . 4 |
25 | unisng 4452 | . . . . 5 | |
26 | 3, 25 | syl 17 | . . . 4 |
27 | 24, 26 | eqtrd 2656 | . . 3 |
28 | 13, 27 | eqtrd 2656 | . 2 |
29 | 10, 28 | eqtrd 2656 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 cdif 3571 wss 3574 cif 4086 csn 4177 cpr 4179 cuni 4436 class class class wbr 4653 cfv 5888 c2o 7554 cen 7952 pmTrspcpmtr 17861 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pmtr 17862 |
This theorem is referenced by: symggen 17890 pmtr3ncomlem1 17893 mdetralt 20414 mdetunilem7 20424 pmtrprfv2 29848 psgnfzto1stlem 29850 pmtridfv1 29857 |
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