fin23lem27 - Metamath Proof Explorer

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Theorem fin23lem27 9150

Description: The mapping constructed in fin23lem22 9149 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)

**Hypothesis**
Ref	Expression
fin23lem22.b

**Assertion**
Ref	Expression
fin23lem27	$/\$

Distinct variable group:

Allowed substitution hints:

(

)

Proof of Theorem fin23lem27 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordom 7074	. . . 4
2		ordwe 5736	. . . 4
3		weso 5105	. . . 4
4	1, 2, 3	mp2b 10	. . 3
5	4	a1i 11	. 2 $/\$
6		sopo 5052	. . . . 5
7	4, 6	ax-mp 5	. . . 4
8		poss 5037	. . . 4
9	7, 8	mpi 20	. . 3
10	9	adantr 481	. 2 $/\$
11		fin23lem22.b	. . . 4
12	11	fin23lem22 9149	. . 3 $/\$
13		f1ofo 6144	. . 3
14	12, 13	syl 17	. 2 $/\$
15		nnsdomel 8816	. . . . . . . 8 $/\$
16	15	adantl 482	. . . . . . 7 $/\$ $/\$ $/\$
17	16	biimpd 219	. . . . . 6 $/\$ $/\$ $/\$
18		fin23lem23 9148	. . . . . . . . . . . . 13 $/\$ $/\$
19	18	adantrr 753	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
20		ineq1 3807	. . . . . . . . . . . . . 14
21	20	breq1d 4663	. . . . . . . . . . . . 13
22	21	cbvreuv 3173	. . . . . . . . . . . 12
23	19, 22	sylib 208	. . . . . . . . . . 11 $/\$ $/\$ $/\$
24		nfv 1843	. . . . . . . . . . . 12
25	21	cbvriotav 6622	. . . . . . . . . . . 12
26		ineq1 3807	. . . . . . . . . . . . 13
27	26	breq1d 4663	. . . . . . . . . . . 12
28	24, 25, 27	riotaprop 6635	. . . . . . . . . . 11 $/\$
29	23, 28	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30	29	simprd 479	. . . . . . . . 9 $/\$ $/\$ $/\$
31	30	adantrr 753	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
32		simprr 796	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
33		fin23lem23 9148	. . . . . . . . . . . . . . 15 $/\$ $/\$
34	33	adantrl 752	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
35	20	breq1d 4663	. . . . . . . . . . . . . . 15
36	35	cbvreuv 3173	. . . . . . . . . . . . . 14
37	34, 36	sylib 208	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
38		nfv 1843	. . . . . . . . . . . . . 14
39	35	cbvriotav 6622	. . . . . . . . . . . . . 14
40		ineq1 3807	. . . . . . . . . . . . . . 15
41	40	breq1d 4663	. . . . . . . . . . . . . 14
42	38, 39, 41	riotaprop 6635	. . . . . . . . . . . . 13 $/\$
43	37, 42	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
44	43	simprd 479	. . . . . . . . . . 11 $/\$ $/\$ $/\$
45	44	ensymd 8007	. . . . . . . . . 10 $/\$ $/\$ $/\$
46	45	adantrr 753	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
47		sdomentr 8094	. . . . . . . . 9 $/\$
48	32, 46, 47	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
49		ensdomtr 8096	. . . . . . . 8 $/\$
50	31, 48, 49	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
51	50	expr 643	. . . . . 6 $/\$ $/\$ $/\$
52		simpll 790	. . . . . . . . 9 $/\$ $/\$ $/\$
53		omsson 7069	. . . . . . . . 9
54	52, 53	syl6ss 3615	. . . . . . . 8 $/\$ $/\$ $/\$
55	29	simpld 475	. . . . . . . 8 $/\$ $/\$ $/\$
56	54, 55	sseldd 3604	. . . . . . 7 $/\$ $/\$ $/\$
57	43	simpld 475	. . . . . . . 8 $/\$ $/\$ $/\$
58	54, 57	sseldd 3604	. . . . . . 7 $/\$ $/\$ $/\$
59		onsdominel 8109	. . . . . . . 8 $/\$ $/\$
60	59	3expia 1267	. . . . . . 7 $/\$
61	56, 58, 60	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$
62	17, 51, 61	3syld 60	. . . . 5 $/\$ $/\$ $/\$
63		simprl 794	. . . . . . 7 $/\$ $/\$ $/\$
64		breq2 4657	. . . . . . . . 9
65	64	riotabidv 6613	. . . . . . . 8
66	65, 11	fvmptg 6280	. . . . . . 7 $/\$
67	63, 55, 66	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$
68		simprr 796	. . . . . . 7 $/\$ $/\$ $/\$
69		breq2 4657	. . . . . . . . 9
70	69	riotabidv 6613	. . . . . . . 8
71	70, 11	fvmptg 6280	. . . . . . 7 $/\$
72	68, 57, 71	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$
73	67, 72	eleq12d 2695	. . . . 5 $/\$ $/\$ $/\$
74	62, 73	sylibrd 249	. . . 4 $/\$ $/\$ $/\$
75		epel 5032	. . . 4
76		fvex 6201	. . . . 5
77	76	epelc 5031	. . . 4
78	74, 75, 77	3imtr4g 285	. . 3 $/\$ $/\$ $/\$
79	78	ralrimivva 2971	. 2 $/\$
80		soisoi 6578	. 2 $/\$ $/\$ $/\$
81	5, 10, 14, 79, 80	syl22anc 1327	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

wreu 2914

cin 3573

wss 3574 class class class wbr 4653

cmpt 4729

cep 5028

wpo 5033

wor 5034

wwe 5072

word 5722

con0 5723

wfo 5886

wf1o 5887

cfv 5888

wiso 5889

crio 6610

com 7065

cen 7952

csdm 7954

cfn 7955

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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

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-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 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-se 5074 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-isom 5897 df-riota 6611 df-om 7066 df-wrecs 7407 df-recs 7468 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765

This theorem is referenced by: (None)

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