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Mirrors > Home > ILE Home > Th. List > oprssdmm | Unicode version |
Description: Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.) |
Ref | Expression |
---|---|
oprssdmm.m | |
oprssdmm.cl | |
oprssdmm.f |
Ref | Expression |
---|---|
oprssdmm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp6 6067 | . . . . . . 7 | |
2 | 1 | biimpi 119 | . . . . . 6 |
3 | 2 | adantl 275 | . . . . 5 |
4 | 3 | simpld 111 | . . . 4 |
5 | 3 | simprd 113 | . . . . 5 |
6 | oprssdmm.f | . . . . . . . . 9 | |
7 | 6 | adantr 274 | . . . . . . . 8 |
8 | eleq2 2203 | . . . . . . . . . 10 | |
9 | 8 | exbidv 1797 | . . . . . . . . 9 |
10 | oprssdmm.m | . . . . . . . . . . 11 | |
11 | 10 | ralrimiva 2505 | . . . . . . . . . 10 |
12 | 11 | adantr 274 | . . . . . . . . 9 |
13 | df-ov 5777 | . . . . . . . . . 10 | |
14 | oprssdmm.cl | . . . . . . . . . 10 | |
15 | 13, 14 | eqeltrrid 2227 | . . . . . . . . 9 |
16 | 9, 12, 15 | rspcdva 2794 | . . . . . . . 8 |
17 | relelfvdm 5453 | . . . . . . . . . 10 | |
18 | 17 | ex 114 | . . . . . . . . 9 |
19 | 18 | exlimdv 1791 | . . . . . . . 8 |
20 | 7, 16, 19 | sylc 62 | . . . . . . 7 |
21 | 20 | ralrimivva 2514 | . . . . . 6 |
22 | 21 | adantr 274 | . . . . 5 |
23 | opeq1 3705 | . . . . . . 7 | |
24 | 23 | eleq1d 2208 | . . . . . 6 |
25 | opeq2 3706 | . . . . . . 7 | |
26 | 25 | eleq1d 2208 | . . . . . 6 |
27 | 24, 26 | rspc2va 2803 | . . . . 5 |
28 | 5, 22, 27 | syl2anc 408 | . . . 4 |
29 | 4, 28 | eqeltrd 2216 | . . 3 |
30 | 29 | ex 114 | . 2 |
31 | 30 | ssrdv 3103 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 wral 2416 wss 3071 cop 3530 cxp 4537 cdm 4539 wrel 4544 cfv 5123 (class class class)co 5774 c1st 6036 c2nd 6037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: axaddf 7676 axmulf 7677 |
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