bcthlem1 - Metamath Proof Explorer

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Theorem bcthlem1 23121

Description: Lemma for bcth 23126. Substitutions for the function

. (Contributed by Mario Carneiro, 9-Jan-2014.)

**Hypotheses**
Ref	Expression
bcth.2
bcthlem.4
bcthlem.5	$/\$ $/\$ $/\$ $\$

**Assertion**
Ref	Expression
bcthlem1	$/\$ $/\$ $/\$ $/\$ $\$

Distinct variable groups:

Allowed substitution hints:

(

)

**Proof of Theorem bcthlem1**
Step	Hyp	Ref	Expression
1		opabssxp 5193	. . . . . . 7 $/\$ $/\$ $/\$ $\$
2		bcthlem.4	. . . . . . . . 9
3		elfvdm 6220	. . . . . . . . 9
4	2, 3	syl 17	. . . . . . . 8
5		reex 10027	. . . . . . . . 9
6		rpssre 11843	. . . . . . . . 9
7	5, 6	ssexi 4803	. . . . . . . 8
8		xpexg 6960	. . . . . . . 8 $/\$
9	4, 7, 8	sylancl 694	. . . . . . 7
10		ssexg 4804	. . . . . . 7 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $/\$ $\$
11	1, 9, 10	sylancr 695	. . . . . 6 $/\$ $/\$ $/\$ $\$
12		oveq2 6658	. . . . . . . . . . 11
13	12	breq2d 4665	. . . . . . . . . 10
14		fveq2 6191	. . . . . . . . . . . 12
15	14	difeq2d 3728	. . . . . . . . . . 11 $\$ $\$
16	15	sseq2d 3633	. . . . . . . . . 10 $\$ $\$
17	13, 16	anbi12d 747	. . . . . . . . 9 $/\$ $\$ $/\$ $\$
18	17	anbi2d 740	. . . . . . . 8 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
19	18	opabbidv 4716	. . . . . . 7 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
20		fveq2 6191	. . . . . . . . . . . 12
21	20	difeq1d 3727	. . . . . . . . . . 11 $\$ $\$
22	21	sseq2d 3633	. . . . . . . . . 10 $\$ $\$
23	22	anbi2d 740	. . . . . . . . 9 $/\$ $\$ $/\$ $\$
24	23	anbi2d 740	. . . . . . . 8 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
25	24	opabbidv 4716	. . . . . . 7 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
26		bcthlem.5	. . . . . . 7 $/\$ $/\$ $/\$ $\$
27	19, 25, 26	ovmpt2g 6795	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
28	11, 27	syl3an3 1361	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
29	28	3expa 1265	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
30	29	ancoms 469	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
31	30	eleq2d 2687	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $\$
32	1	sseli 3599	. . 3 $/\$ $/\$ $/\$ $\$
33		simp1 1061	. . 3 $/\$ $/\$ $\$
34		1st2nd2 7205	. . . . . 6
35	34	eleq1d 2686	. . . . 5 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
36		fvex 6201	. . . . . 6
37		fvex 6201	. . . . . 6
38		eleq1 2689	. . . . . . . 8
39		eleq1 2689	. . . . . . . 8
40	38, 39	bi2anan9 917	. . . . . . 7 $/\$ $/\$ $/\$
41		simpr 477	. . . . . . . . 9 $/\$
42	41	breq1d 4663	. . . . . . . 8 $/\$
43		oveq12 6659	. . . . . . . . . 10 $/\$
44	43	fveq2d 6195	. . . . . . . . 9 $/\$
45	44	sseq1d 3632	. . . . . . . 8 $/\$ $\$ $\$
46	42, 45	anbi12d 747	. . . . . . 7 $/\$ $/\$ $\$ $/\$ $\$
47	40, 46	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
48	36, 37, 47	opelopaba 4991	. . . . 5 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
49	35, 48	syl6bb 276	. . . 4 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $/\$ $\$
50	34	eleq1d 2686	. . . . . . 7
51		opelxp 5146	. . . . . . 7 $/\$
52	50, 51	syl6rbb 277	. . . . . 6 $/\$
53	34	fveq2d 6195	. . . . . . . . . 10
54		df-ov 6653	. . . . . . . . . 10
55	53, 54	syl6reqr 2675	. . . . . . . . 9
56	55	fveq2d 6195	. . . . . . . 8
57	56	sseq1d 3632	. . . . . . 7 $\$ $\$
58	57	anbi2d 740	. . . . . 6 $/\$ $\$ $/\$ $\$
59	52, 58	anbi12d 747	. . . . 5 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $\$
60		3anass 1042	. . . . 5 $/\$ $/\$ $\$ $/\$ $/\$ $\$
61	59, 60	syl6bbr 278	. . . 4 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $\$
62	49, 61	bitrd 268	. . 3 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $\$
63	32, 33, 62	pm5.21nii 368	. 2 $/\$ $/\$ $/\$ $\$ $/\$ $/\$ $\$
64	31, 63	syl6bb 276	1 $/\$ $/\$ $/\$ $/\$ $\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990

cvv 3200 $\$ cdif 3571

wss 3574

cop 4183 class class class wbr 4653

copab 4712

cxp 5112

cdm 5114

cfv 5888 (class class class)co 6650

cmpt2 6652

c1st 7166

c2nd 7167

cr 9935

c1 9937

clt 10074

cdiv 10684

cn 11020

crp 11832

cbl 19733

cmopn 19736

ccl 20822

cms 23052

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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-rp 11833

This theorem is referenced by: bcthlem2 23122 bcthlem3 23123 bcthlem4 23124

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