eldioph2lem2 - Mathbox for Stefan O'Rear

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Theorem eldioph2lem2 35603

Description: Lemma for eldioph2 35604. 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 762	. . . 4 $/\$ $/\$ $/\$
2		fzfi 12185	. . . 4
3		difinf 7841	. . . 4 $/\$ $\$
4	1, 2, 3	sylancl 668	. . 3 $/\$ $/\$ $/\$ $\$
5		fzfi 12185	. . . 4
6		diffi 7803	. . . 4 $\$
7	5, 6	ax-mp 5	. . 3 $\$
8		isinffi 8426	. . 3 $\$ $/\$ $\$ $\$ $\$
9	4, 7, 8	sylancl 668	. 2 $/\$ $/\$ $/\$ $\$ $\$
10		f1f1orn 5825	. . . . . . . 8 $\$ $\$ $\$
11	10	adantl 468	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
12		f1oi 5850	. . . . . . . 8
13	12	a1i 11	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$
14		incom 3625	. . . . . . . . 9 $\$ $\$
15		disjdif 3839	. . . . . . . . 9 $\$
16	14, 15	eqtri 2473	. . . . . . . 8 $\$
17	16	a1i 11	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
18		f1f 5779	. . . . . . . . . . . 12 $\$ $\$ $\$ $\$
19		frn 5735	. . . . . . . . . . . 12 $\$ $\$ $\$
20	18, 19	syl 17	. . . . . . . . . . 11 $\$ $\$ $\$
21	20	adantl 468	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
22		ssrin 3657	. . . . . . . . . 10 $\$ $\$
23	21, 22	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
24		incom 3625	. . . . . . . . . 10 $\$ $\$
25		disjdif 3839	. . . . . . . . . 10 $\$
26	24, 25	eqtri 2473	. . . . . . . . 9 $\$
27	23, 26	syl6sseq 3478	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $\$ $\$
28		ss0 3765	. . . . . . . 8
29	27, 28	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$
30		f1oun 5833	. . . . . . 7 $\$ $/\$ $/\$ $\$ $/\$ $\$
31	11, 13, 17, 29, 30	syl22anc 1269	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
32		f1of1 5813	. . . . . 6 $\$ $\$
33	31, 32	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
34		uncom 3578	. . . . . . 7 $\$ $\$
35		simplrr 771	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $\$ $\$
36		fzss2 11838	. . . . . . . . 9
37	35, 36	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $\$ $\$
38		undif 3848	. . . . . . . 8 $\$
39	37, 38	sylib 200	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
40	34, 39	syl5eq 2497	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
41		f1eq2 5775	. . . . . 6 $\$ $\$
42	40, 41	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
43	33, 42	mpbid 214	. . . 4 $/\$ $/\$ $/\$ $/\$ $\$ $\$
44	20	difss2d 3563	. . . . . 6 $\$ $\$
45	44	adantl 468	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$
46		simplrl 770	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$
47	45, 46	unssd 3610	. . . 4 $/\$ $/\$ $/\$ $/\$ $\$ $\$
48		f1ss 5784	. . . 4 $/\$
49	43, 47, 48	syl2anc 667	. . 3 $/\$ $/\$ $/\$ $/\$ $\$ $\$
50		resundir 5119	. . . 4
51		dmres 5125	. . . . . . . 8
52		incom 3625	. . . . . . . . 9
53		f1dm 5783	. . . . . . . . . . . 12 $\$ $\$ $\$
54	53	adantl 468	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
55	54	ineq1d 3633	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $\$
56	55, 16	syl6eq 2501	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $\$ $\$
57	52, 56	syl5eq 2497	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $\$ $\$
58	51, 57	syl5eq 2497	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $\$ $\$
59		relres 5132	. . . . . . . 8
60		reldm0 5052	. . . . . . . 8
61	59, 60	ax-mp 5	. . . . . . 7
62	58, 61	sylibr 216	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\$ $\$
63		residm 5136	. . . . . . 7
64	63	a1i 11	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\$ $\$
65	62, 64	uneq12d 3589	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\$ $\$
66		uncom 3578	. . . . . 6
67		un0 3759	. . . . . 6
68	66, 67	eqtri 2473	. . . . 5
69	65, 68	syl6eq 2501	. . . 4 $/\$ $/\$ $/\$ $/\$ $\$ $\$
70	50, 69	syl5eq 2497	. . 3 $/\$ $/\$ $/\$ $/\$ $\$ $\$
71		vex 3048	. . . . 5
72		ovex 6318	. . . . . 6
73		resiexg 6729	. . . . . 6
74	72, 73	ax-mp 5	. . . . 5
75	71, 74	unex 6589	. . . 4
76		f1eq1 5774	. . . . 5
77		reseq1 5099	. . . . . 6
78	77	eqeq1d 2453	. . . . 5
79	76, 78	anbi12d 717	. . . 4 $/\$ $/\$
80	75, 79	spcev 3141	. . 3 $/\$ $/\$
81	49, 70, 80	syl2anc 667	. 2 $/\$ $/\$ $/\$ $/\$ $\$ $\$ $/\$
82	9, 81	exlimddv 1781	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 188 $/\$ wa 371

wceq 1444

wex 1663

wcel 1887

cvv 3045 $\$ cdif 3401

cun 3402

cin 3403

wss 3404

c0 3731

cid 4744

cdm 4834

crn 4835

cres 4836

wrel 4839

wf 5578

wf1 5579

wf1o 5581

cfv 5582 (class class class)co 6290

cfn 7569

c1 9540

cn0 10869

cuz 11159

cfz 11784

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616

This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-oadd 7186 df-er 7363 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-card 8373 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-nn 10610 df-n0 10870 df-z 10938 df-uz 11160 df-fz 11785

This theorem is referenced by: eldioph2b 35605

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