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Mirrors > Home > ILE Home > Th. List > isstruct2r | Unicode version |
Description: The property of being a structure with components in . (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
Ref | Expression |
---|---|
isstruct2r | Struct |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 518 | . 2 | |
2 | simplr 519 | . 2 | |
3 | simprr 521 | . 2 | |
4 | simprl 520 | . . . 4 | |
5 | 4 | elexd 2694 | . . 3 |
6 | elex 2692 | . . . 4 | |
7 | 6 | ad2antrr 479 | . . 3 |
8 | simpr 109 | . . . . . 6 | |
9 | 8 | eleq1d 2206 | . . . . 5 |
10 | simpl 108 | . . . . . . 7 | |
11 | 10 | difeq1d 3188 | . . . . . 6 |
12 | 11 | funeqd 5140 | . . . . 5 |
13 | 10 | dmeqd 4736 | . . . . . 6 |
14 | 8 | fveq2d 5418 | . . . . . 6 |
15 | 13, 14 | sseq12d 3123 | . . . . 5 |
16 | 9, 12, 15 | 3anbi123d 1290 | . . . 4 |
17 | df-struct 11950 | . . . 4 Struct | |
18 | 16, 17 | brabga 4181 | . . 3 Struct |
19 | 5, 7, 18 | syl2anc 408 | . 2 Struct |
20 | 1, 2, 3, 19 | mpbir3and 1164 | 1 Struct |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 cvv 2681 cdif 3063 cin 3065 wss 3066 c0 3358 csn 3522 class class class wbr 3924 cxp 4532 cdm 4534 wfun 5112 cfv 5118 cle 7794 cn 8713 cfz 9783 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-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-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 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: isstructr 11963 |
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