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Mirrors > Home > MPE Home > Th. List > relexp0g | Structured version Visualization version Unicode version |
Description: A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.) |
Ref | Expression |
---|---|
relexp0g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2623 | . . 3 | |
2 | simprr 796 | . . . . 5 | |
3 | 2 | iftrued 4094 | . . . 4 |
4 | dmeq 5324 | . . . . . . 7 | |
5 | rneq 5351 | . . . . . . 7 | |
6 | 4, 5 | uneq12d 3768 | . . . . . 6 |
7 | 6 | reseq2d 5396 | . . . . 5 |
8 | 7 | ad2antrl 764 | . . . 4 |
9 | 3, 8 | eqtrd 2656 | . . 3 |
10 | elex 3212 | . . 3 | |
11 | 0nn0 11307 | . . . 4 | |
12 | 11 | a1i 11 | . . 3 |
13 | dmexg 7097 | . . . . 5 | |
14 | rnexg 7098 | . . . . 5 | |
15 | unexg 6959 | . . . . 5 | |
16 | 13, 14, 15 | syl2anc 693 | . . . 4 |
17 | resiexg 7102 | . . . 4 | |
18 | 16, 17 | syl 17 | . . 3 |
19 | 1, 9, 10, 12, 18 | ovmpt2d 6788 | . 2 |
20 | df-relexp 13761 | . . 3 | |
21 | oveq 6656 | . . . . 5 | |
22 | 21 | eqeq1d 2624 | . . . 4 |
23 | 22 | imbi2d 330 | . . 3 |
24 | 20, 23 | ax-mp 5 | . 2 |
25 | 19, 24 | mpbir 221 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cvv 3200 cun 3572 cif 4086 cmpt 4729 cid 5023 cdm 5114 crn 5115 cres 5116 ccom 5118 cfv 5888 (class class class)co 6650 cmpt2 6652 cc0 9936 c1 9937 cn0 11292 cseq 12801 crelexp 13760 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-i2m1 10004 |
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-pw 4160 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-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-n0 11293 df-relexp 13761 |
This theorem is referenced by: relexp0 13763 relexpcnv 13775 relexp0rel 13777 relexpdmg 13782 relexprng 13786 relexpfld 13789 relexpaddg 13793 dfrcl3 37967 fvmptiunrelexplb0d 37976 brfvrcld2 37984 relexp0eq 37993 iunrelexp0 37994 relexpiidm 37996 relexpss1d 37997 relexpmulg 38002 iunrelexpmin2 38004 relexp01min 38005 relexp0a 38008 relexpxpmin 38009 relexpaddss 38010 dfrtrcl3 38025 cotrclrcl 38034 |
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