eldioph2lem2 - Mathbox for Stefan O'Rear

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Theorem eldioph2lem2 35674

Description: Lemma for eldioph2 35675. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)

**Assertion**
Ref	Expression
eldioph2lem2	$/\$ $/\$ $/\$ $/\$

Distinct variable groups:

Proof of Theorem eldioph2lem2 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 770	. . . 4 $/\$ $/\$ $/\$
2		fzfi 12223	. . . 4
3		difinf 7859	. . . 4 $/\$ $\$
4	1, 2, 3	sylancl 675	. . 3 $/\$ $/\$ $/\$ $\$
5		fzfi 12223	. . . 4
6		diffi 7821	. . . 4 $\$
7	5, 6	ax-mp 5	. . 3 $\$
8		isinffi 8444	. . 3 $\$ $/\$ $\$ $\$ $\$
9	4, 7, 8	sylancl 675	. 2 $/\$ $/\$ $/\$ $\$ $\$
10		f1f1orn 5839	. . . . . . . 8 $\$ $\$ $\$
11	10	adantl 473	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
12		f1oi 5864	. . . . . . . 8
13	12	a1i 11	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$
14		incom 3616	. . . . . . . . 9 $\$ $\$
15		disjdif 3830	. . . . . . . . 9 $\$
16	14, 15	eqtri 2493	. . . . . . . 8 $\$
17	16	a1i 11	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
18		f1f 5792	. . . . . . . . . . . 12 $\$ $\$ $\$ $\$
19		frn 5747	. . . . . . . . . . . 12 $\$ $\$ $\$
20	18, 19	syl 17	. . . . . . . . . . 11 $\$ $\$ $\$
21	20	adantl 473	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
22		ssrin 3648	. . . . . . . . . 10 $\$ $\$
23	21, 22	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
24		incom 3616	. . . . . . . . . 10 $\$ $\$
25		disjdif 3830	. . . . . . . . . 10 $\$
26	24, 25	eqtri 2493	. . . . . . . . 9 $\$
27	23, 26	syl6sseq 3464	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $\$ $\$
28		ss0 3768	. . . . . . . 8
29	27, 28	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$
30		f1oun 5847	. . . . . . 7 $\$ $/\$ $/\$ $\$ $/\$ $\$
31	11, 13, 17, 29, 30	syl22anc 1293	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
32		f1of1 5827	. . . . . 6 $\$ $\$
33	31, 32	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
34		uncom 3569	. . . . . . 7 $\$ $\$
35		simplrr 779	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $\$ $\$
36		fzss2 11864	. . . . . . . . 9
37	35, 36	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $\$ $\$
38		undif 3839	. . . . . . . 8 $\$
39	37, 38	sylib 201	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
40	34, 39	syl5eq 2517	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
41		f1eq2 5788	. . . . . 6 $\$ $\$
42	40, 41	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
43	33, 42	mpbid 215	. . . 4 $/\$ $/\$ $/\$ $/\$ $\$ $\$
44	20	difss2d 3552	. . . . . 6 $\$ $\$
45	44	adantl 473	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$
46		simplrl 778	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$
47	45, 46	unssd 3601	. . . 4 $/\$ $/\$ $/\$ $/\$ $\$ $\$
48		f1ss 5797	. . . 4 $/\$
49	43, 47, 48	syl2anc 673	. . 3 $/\$ $/\$ $/\$ $/\$ $\$ $\$
50		resundir 5125	. . . 4
51		dmres 5131	. . . . . . . 8
52		incom 3616	. . . . . . . . 9
53		f1dm 5796	. . . . . . . . . . . 12 $\$ $\$ $\$
54	53	adantl 473	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
55	54	ineq1d 3624	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
56	55, 16	syl6eq 2521	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $\$ $\$
57	52, 56	syl5eq 2517	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $\$ $\$
58	51, 57	syl5eq 2517	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$
59		relres 5138	. . . . . . . 8
60		reldm0 5058	. . . . . . . 8
61	59, 60	ax-mp 5	. . . . . . 7
62	58, 61	sylibr 217	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\$ $\$
63		residm 5142	. . . . . . 7
64	63	a1i 11	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\$ $\$
65	62, 64	uneq12d 3580	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$
66		uncom 3569	. . . . . 6
67		un0 3762	. . . . . 6
68	66, 67	eqtri 2493	. . . . 5
69	65, 68	syl6eq 2521	. . . 4 $/\$ $/\$ $/\$ $/\$ $\$ $\$
70	50, 69	syl5eq 2517	. . 3 $/\$ $/\$ $/\$ $/\$ $\$ $\$
71		vex 3034	. . . . 5
72		ovex 6336	. . . . . 6
73		resiexg 6748	. . . . . 6
74	72, 73	ax-mp 5	. . . . 5
75	71, 74	unex 6608	. . . 4
76		f1eq1 5787	. . . . 5
77		reseq1 5105	. . . . . 6
78	77	eqeq1d 2473	. . . . 5
79	76, 78	anbi12d 725	. . . 4 $/\$ $/\$
80	75, 79	spcev 3127	. . 3 $/\$ $/\$
81	49, 70, 80	syl2anc 673	. 2 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $/\$
82	9, 81	exlimddv 1789	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 189 $/\$ wa 376

wceq 1452

wex 1671

wcel 1904

cvv 3031 $\$ cdif 3387

cun 3388

cin 3389

wss 3390

c0 3722

cid 4749

cdm 4839

crn 4840

cres 4841

wrel 4844

wf 5585

wf1 5586

wf1o 5588

cfv 5589 (class class class)co 6308

cfn 7587

c1 9558

cn0 10893

cuz 11182

cfz 11810

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-cnex 9613 ax-resscn 9614 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-addrcl 9618 ax-mulcl 9619 ax-mulrcl 9620 ax-mulcom 9621 ax-addass 9622 ax-mulass 9623 ax-distr 9624 ax-i2m1 9625 ax-1ne0 9626 ax-1rid 9627 ax-rnegex 9628 ax-rrecex 9629 ax-cnre 9630 ax-pre-lttri 9631 ax-pre-lttrn 9632 ax-pre-ltadd 9633 ax-pre-mulgt0 9634

This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-int 4227 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-pred 5387 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-om 6712 df-1st 6812 df-2nd 6813 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-1o 7200 df-oadd 7204 df-er 7381 df-en 7588 df-dom 7589 df-sdom 7590 df-fin 7591 df-card 8391 df-pnf 9695 df-mnf 9696 df-xr 9697 df-ltxr 9698 df-le 9699 df-sub 9882 df-neg 9883 df-nn 10632 df-n0 10894 df-z 10962 df-uz 11183 df-fz 11811

This theorem is referenced by: eldioph2b 35676

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