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Mirrors > Home > ILE Home > Th. List > structcnvcnv | Unicode version |
Description: Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
structcnvcnv | Struct |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 4562 | . . . . . 6 | |
2 | cnvcnv 4986 | . . . . . . . 8 | |
3 | inss2 3292 | . . . . . . . 8 | |
4 | 2, 3 | eqsstri 3124 | . . . . . . 7 |
5 | 4 | sseli 3088 | . . . . . 6 |
6 | 1, 5 | mto 651 | . . . . 5 |
7 | disjsn 3580 | . . . . 5 | |
8 | 6, 7 | mpbir 145 | . . . 4 |
9 | cnvcnvss 4988 | . . . . 5 | |
10 | reldisj 3409 | . . . . 5 | |
11 | 9, 10 | ax-mp 5 | . . . 4 |
12 | 8, 11 | mpbi 144 | . . 3 |
13 | 12 | a1i 9 | . 2 Struct |
14 | structn0fun 11961 | . . . . 5 Struct | |
15 | funrel 5135 | . . . . 5 | |
16 | 14, 15 | syl 14 | . . . 4 Struct |
17 | dfrel2 4984 | . . . 4 | |
18 | 16, 17 | sylib 121 | . . 3 Struct |
19 | difss 3197 | . . . 4 | |
20 | cnvss 4707 | . . . 4 | |
21 | cnvss 4707 | . . . 4 | |
22 | 19, 20, 21 | mp2b 8 | . . 3 |
23 | 18, 22 | eqsstrrdi 3145 | . 2 Struct |
24 | 13, 23 | eqssd 3109 | 1 Struct |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wceq 1331 wcel 1480 cvv 2681 cdif 3063 cin 3065 wss 3066 c0 3358 csn 3522 class class class wbr 3924 cxp 4532 ccnv 4533 wrel 4539 wfun 5112 Struct cstr 11944 |
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-in1 603 ax-in2 604 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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-struct 11950 |
This theorem is referenced by: structfung 11965 |
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