mapunen - Metamath Proof Explorer

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Theorem mapunen 8129

Description: Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

**Assertion**
Ref	Expression
mapunen	$/\$ $/\$ $/\$

Proof of Theorem mapunen Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3
2	1	a1i 11	. 2 $/\$ $/\$ $/\$
3		ovex 6678	. . . 4
4		ovex 6678	. . . 4
5	3, 4	xpex 6962	. . 3
6	5	a1i 11	. 2 $/\$ $/\$ $/\$
7		elmapi 7879	. . . . 5
8		ssun1 3776	. . . . 5
9		fssres 6070	. . . . 5 $/\$
10	7, 8, 9	sylancl 694	. . . 4
11		ssun2 3777	. . . . 5
12		fssres 6070	. . . . 5 $/\$
13	7, 11, 12	sylancl 694	. . . 4
14	10, 13	jca 554	. . 3 $/\$
15		opelxp 5146	. . . 4 $/\$
16		simpl3 1066	. . . . . 6 $/\$ $/\$ $/\$
17		simpl1 1064	. . . . . 6 $/\$ $/\$ $/\$
18	16, 17	elmapd 7871	. . . . 5 $/\$ $/\$ $/\$
19		simpl2 1065	. . . . . 6 $/\$ $/\$ $/\$
20	16, 19	elmapd 7871	. . . . 5 $/\$ $/\$ $/\$
21	18, 20	anbi12d 747	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
22	15, 21	syl5bb 272	. . 3 $/\$ $/\$ $/\$ $/\$
23	14, 22	syl5ibr 236	. 2 $/\$ $/\$ $/\$
24		xp1st 7198	. . . . . . 7
25	24	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$
26		elmapi 7879	. . . . . 6
27	25, 26	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$
28		xp2nd 7199	. . . . . . 7
29	28	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$
30		elmapi 7879	. . . . . 6
31	29, 30	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$
32		simplr 792	. . . . 5 $/\$ $/\$ $/\$ $/\$
33		fun2 6067	. . . . 5 $/\$ $/\$
34	27, 31, 32, 33	syl21anc 1325	. . . 4 $/\$ $/\$ $/\$ $/\$
35	34	ex 450	. . 3 $/\$ $/\$ $/\$
36		unexg 6959	. . . . 5 $/\$
37	17, 19, 36	syl2anc 693	. . . 4 $/\$ $/\$ $/\$
38	16, 37	elmapd 7871	. . 3 $/\$ $/\$ $/\$
39	35, 38	sylibrd 249	. 2 $/\$ $/\$ $/\$
40		1st2nd2 7205	. . . . . . 7
41	40	ad2antll 765	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
42	27	adantrl 752	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
43	31	adantrl 752	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44		res0 5400	. . . . . . . . . 10
45		res0 5400	. . . . . . . . . 10
46	44, 45	eqtr4i 2647	. . . . . . . . 9
47		simplr 792	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
48	47	reseq2d 5396	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
49	47	reseq2d 5396	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
50	46, 48, 49	3eqtr4a 2682	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
51		fresaunres1 6077	. . . . . . . 8 $/\$ $/\$
52	42, 43, 50, 51	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
53		fresaunres2 6076	. . . . . . . 8 $/\$ $/\$
54	42, 43, 50, 53	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
55	52, 54	opeq12d 4410	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
56	41, 55	eqtr4d 2659	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
57		reseq1 5390	. . . . . . 7
58		reseq1 5390	. . . . . . 7
59	57, 58	opeq12d 4410	. . . . . 6
60	59	eqeq2d 2632	. . . . 5
61	56, 60	syl5ibrcom 237	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
62		ffn 6045	. . . . . . . 8
63		fnresdm 6000	. . . . . . . 8
64	7, 62, 63	3syl 18	. . . . . . 7
65	64	ad2antrl 764	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
66	65	eqcomd 2628	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
67		vex 3203	. . . . . . . . . 10
68	67	resex 5443	. . . . . . . . 9
69	67	resex 5443	. . . . . . . . 9
70	68, 69	op1std 7178	. . . . . . . 8
71	68, 69	op2ndd 7179	. . . . . . . 8
72	70, 71	uneq12d 3768	. . . . . . 7
73		resundi 5410	. . . . . . 7
74	72, 73	syl6eqr 2674	. . . . . 6
75	74	eqeq2d 2632	. . . . 5
76	66, 75	syl5ibrcom 237	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	61, 76	impbid 202	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
78	77	ex 450	. 2 $/\$ $/\$ $/\$ $/\$
79	2, 6, 23, 39, 78	en3d 7992	1 $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990

cvv 3200

cun 3572

cin 3573

wss 3574

c0 3915

cop 4183 class class class wbr 4653

cxp 5112

cres 5116

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

cmap 7857

cen 7952

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

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-csb 3534 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-iun 4522 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-res 5126 df-ima 5127 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-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-en 7956

This theorem is referenced by: map2xp 8130 mapdom2 8131 mapcdaen 9006 ackbij1lem5 9046 hashmap 13222 mpct 39393

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