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Theorem setsstruct2 15896

Description: An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.)

**Assertion**
Ref	Expression
setsstruct2	Struct $/\$ $/\$ $/\$ sSet Struct

**Proof of Theorem setsstruct2**
Step	Hyp	Ref	Expression
1		isstruct2 15867	. . . . . . 7 Struct $/\$ $\$ $/\$
2		elin 3796	. . . . . . . . 9 $/\$
3		elxp6 7200	. . . . . . . . . . 11 $/\$ $/\$
4		eleq1 2689	. . . . . . . . . . . . 13
5	4	adantr 481	. . . . . . . . . . . 12 $/\$ $/\$
6		simp3 1063	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
7		simp1l 1085	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
8	6, 7	ifcld 4131	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
9	8	nnred 11035	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
10	6	nnred 11035	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
11		simp1r 1086	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
12	11, 6	ifcld 4131	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$
13	12	nnred 11035	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
14		nnre 11027	. . . . . . . . . . . . . . . . . . . . . 22
15	14	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 $/\$
16		nnre 11027	. . . . . . . . . . . . . . . . . . . . 21
17	15, 16	anim12i 590	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$
18	17	3adant2 1080	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$
19	18	ancomd 467	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
20		min1 12020	. . . . . . . . . . . . . . . . . 18 $/\$
21	19, 20	syl 17	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
22		nnre 11027	. . . . . . . . . . . . . . . . . . . . . 22
23	22	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 $/\$
24	23, 16	anim12i 590	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$
25	24	3adant2 1080	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$
26	25	ancomd 467	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
27		max1 12016	. . . . . . . . . . . . . . . . . 18 $/\$
28	26, 27	syl 17	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
29	9, 10, 13, 21, 28	letrd 10194	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
30		df-br 4654	. . . . . . . . . . . . . . . 16
31	29, 30	sylib 208	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
32	8, 12	opelxpd 5149	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
33	31, 32	elind 3798	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
34	33	3exp 1264	. . . . . . . . . . . . 13 $/\$
35	34	adantl 482	. . . . . . . . . . . 12 $/\$ $/\$
36	5, 35	sylbid 230	. . . . . . . . . . 11 $/\$ $/\$
37	3, 36	sylbi 207	. . . . . . . . . 10
38	37	impcom 446	. . . . . . . . 9 $/\$
39	2, 38	sylbi 207	. . . . . . . 8
40	39	3ad2ant1 1082	. . . . . . 7 $/\$ $\$ $/\$
41	1, 40	sylbi 207	. . . . . 6 Struct
42	41	imp 445	. . . . 5 Struct $/\$
43	42	3adant2 1080	. . . 4 Struct $/\$ $/\$
44		structex 15868	. . . . . . 7 Struct
45		structn0fun 15869	. . . . . . 7 Struct $\$
46	44, 45	jca 554	. . . . . 6 Struct $/\$ $\$
47	46	3ad2ant1 1082	. . . . 5 Struct $/\$ $/\$ $/\$ $\$
48		simp3 1063	. . . . 5 Struct $/\$ $/\$
49		simp2 1062	. . . . 5 Struct $/\$ $/\$
50		setsfun0 15894	. . . . 5 $/\$ $\$ $/\$ $/\$ sSet $\$
51	47, 48, 49, 50	syl12anc 1324	. . . 4 Struct $/\$ $/\$ sSet $\$
52	44	3ad2ant1 1082	. . . . . . 7 Struct $/\$ $/\$
53	52, 49	jca 554	. . . . . 6 Struct $/\$ $/\$ $/\$
54		setsdm 15892	. . . . . 6 $/\$ sSet
55	53, 54	syl 17	. . . . 5 Struct $/\$ $/\$ sSet
56		fveq2 6191	. . . . . . . . . . . . . . . . 17
57		df-ov 6653	. . . . . . . . . . . . . . . . 17
58	56, 57	syl6eqr 2674	. . . . . . . . . . . . . . . 16
59	58	sseq2d 3633	. . . . . . . . . . . . . . 15
60	59	adantr 481	. . . . . . . . . . . . . 14 $/\$ $/\$
61		df-3an 1039	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
62		nnz 11399	. . . . . . . . . . . . . . . . . . . . 21
63		nnz 11399	. . . . . . . . . . . . . . . . . . . . 21
64		nnz 11399	. . . . . . . . . . . . . . . . . . . . 21
65	62, 63, 64	3anim123i 1247	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$
66		ssfzunsnext 12386	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$
67		df-ov 6653	. . . . . . . . . . . . . . . . . . . . 21
68	66, 67	syl6sseq 3651	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$
69	65, 68	sylan2 491	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
70	69	ex 450	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
71	61, 70	syl5bir 233	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
72	71	expd 452	. . . . . . . . . . . . . . . 16 $/\$
73	72	com12 32	. . . . . . . . . . . . . . 15 $/\$
74	73	adantl 482	. . . . . . . . . . . . . 14 $/\$ $/\$
75	60, 74	sylbid 230	. . . . . . . . . . . . 13 $/\$ $/\$
76	3, 75	sylbi 207	. . . . . . . . . . . 12
77	76	adantl 482	. . . . . . . . . . 11 $/\$
78	2, 77	sylbi 207	. . . . . . . . . 10
79	78	imp 445	. . . . . . . . 9 $/\$
80	79	3adant2 1080	. . . . . . . 8 $/\$ $\$ $/\$
81	1, 80	sylbi 207	. . . . . . 7 Struct
82	81	imp 445	. . . . . 6 Struct $/\$
83	82	3adant2 1080	. . . . 5 Struct $/\$ $/\$
84	55, 83	eqsstrd 3639	. . . 4 Struct $/\$ $/\$ sSet
85		isstruct2 15867	. . . 4 sSet Struct $/\$ sSet $\$ $/\$ sSet
86	43, 51, 84, 85	syl3anbrc 1246	. . 3 Struct $/\$ $/\$ sSet Struct
87	86	adantr 481	. 2 Struct $/\$ $/\$ $/\$ sSet Struct
88		breq2 4657	. . 3 sSet Struct sSet Struct
89	88	adantl 482	. 2 Struct $/\$ $/\$ $/\$ sSet Struct sSet Struct
90	87, 89	mpbird 247	1 Struct $/\$ $/\$ $/\$ sSet Struct

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cvv 3200 $\$ cdif 3571

cun 3572

cin 3573

wss 3574

c0 3915

cif 4086

csn 4177

cop 4183 class class class wbr 4653

cxp 5112

cdm 5114

wfun 5882

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

cr 9935

cle 10075

cn 11020

cz 11377

cfz 12326 Struct cstr 15853 sSet csts 15855

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-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-sets 15864

This theorem is referenced by: setsexstruct2 15897 setsstruct 15898

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