Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pmtrfval | Structured version Visualization version Unicode version |
Description: The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
Ref | Expression |
---|---|
pmtrfval.t | pmTrsp |
Ref | Expression |
---|---|
pmtrfval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrfval.t | . 2 pmTrsp | |
2 | elex 3212 | . . 3 | |
3 | pweq 4161 | . . . . . 6 | |
4 | 3 | rabeqdv 3194 | . . . . 5 |
5 | mpteq1 4737 | . . . . 5 | |
6 | 4, 5 | mpteq12dv 4733 | . . . 4 |
7 | df-pmtr 17862 | . . . 4 pmTrsp | |
8 | vpwex 4849 | . . . . 5 | |
9 | 8 | mptrabex 6488 | . . . 4 |
10 | 6, 7, 9 | fvmpt3i 6287 | . . 3 pmTrsp |
11 | 2, 10 | syl 17 | . 2 pmTrsp |
12 | 1, 11 | syl5eq 2668 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 crab 2916 cvv 3200 cdif 3571 cif 4086 cpw 4158 csn 4177 cuni 4436 class class class wbr 4653 cmpt 4729 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-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 |
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-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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-pmtr 17862 |
This theorem is referenced by: pmtrval 17871 pmtrrn 17877 pmtrfrn 17878 pmtrprfval 17907 pmtrsn 17939 |
Copyright terms: Public domain | W3C validator |