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Mirrors > Home > MPE Home > Th. List > coires1 | Structured version Visualization version Unicode version |
Description: Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
coires1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cocnvcnv1 5646 | . . . . 5 | |
2 | relcnv 5503 | . . . . . 6 | |
3 | coi1 5651 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | 1, 4 | eqtr3i 2646 | . . . 4 |
6 | 5 | reseq1i 5392 | . . 3 |
7 | resco 5639 | . . 3 | |
8 | 6, 7 | eqtr3i 2646 | . 2 |
9 | rescnvcnv 5597 | . 2 | |
10 | 8, 9 | eqtr3i 2646 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 cid 5023 ccnv 5113 cres 5116 ccom 5118 wrel 5119 |
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-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-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 |
This theorem is referenced by: relcoi1 5664 funcoeqres 6167 relexpaddg 13793 psrass1lem 19377 lindfres 20162 lindsmm 20167 kgencn2 21360 ustssco 22018 erdsze2lem2 31186 poimirlem9 33418 mzpresrename 37313 diophrw 37322 eldioph2 37325 diophren 37377 relexpiidm 37996 relexpaddss 38010 cotrclrcl 38034 funcrngcsetcALT 41999 |
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