fntpb - Metamath Proof Explorer

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Theorem fntpb 6473

Description: A function whose domain has at most three elements can be represented as a set of at most three ordered pairs. (Contributed by AV, 26-Jan-2021.)

**Hypotheses**
Ref	Expression
fnprb.a
fnprb.b
fntpb.c

**Assertion**
Ref	Expression
fntpb

Proof of Theorem fntpb Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnprb.a	. . . . . . . 8
2		fnprb.b	. . . . . . . 8
3	1, 2	fnprb 6472	. . . . . . 7
4		tpidm23 4292	. . . . . . . . 9
5	4	eqcomi 2631	. . . . . . . 8
6	5	fneq2i 5986	. . . . . . 7
7		tpidm23 4292	. . . . . . . . 9
8	7	eqcomi 2631	. . . . . . . 8
9	8	eqeq2i 2634	. . . . . . 7
10	3, 6, 9	3bitr3i 290	. . . . . 6
11	10	a1i 11	. . . . 5
12		tpeq3 4279	. . . . . 6
13	12	fneq2d 5982	. . . . 5
14		id 22	. . . . . . . 8
15		fveq2 6191	. . . . . . . 8
16	14, 15	opeq12d 4410	. . . . . . 7
17	16	tpeq3d 4282	. . . . . 6
18	17	eqeq2d 2632	. . . . 5
19	11, 13, 18	3bitr3d 298	. . . 4
20	19	a1d 25	. . 3 $/\$
21		fndm 5990	. . . . . . . 8
22		fvex 6201	. . . . . . . . 9
23		fvex 6201	. . . . . . . . 9
24		fvex 6201	. . . . . . . . 9
25	22, 23, 24	dmtpop 5611	. . . . . . . 8
26	21, 25	syl6eqr 2674	. . . . . . 7
27	26	adantl 482	. . . . . 6 $/\$ $/\$ $/\$
28	21	adantl 482	. . . . . . . . 9 $/\$ $/\$ $/\$
29	28	eleq2d 2687	. . . . . . . 8 $/\$ $/\$ $/\$
30		vex 3203	. . . . . . . . . 10
31	30	eltp 4230	. . . . . . . . 9 $\/$ $\/$
32	1, 22	fvtp1 6460	. . . . . . . . . . . . 13 $/\$
33	32	ad2antrr 762	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
34	33	eqcomd 2628	. . . . . . . . . . 11 $/\$ $/\$ $/\$
35		fveq2 6191	. . . . . . . . . . . 12
36		fveq2 6191	. . . . . . . . . . . 12
37	35, 36	eqeq12d 2637	. . . . . . . . . . 11
38	34, 37	syl5ibrcom 237	. . . . . . . . . 10 $/\$ $/\$ $/\$
39	2, 23	fvtp2 6461	. . . . . . . . . . . . 13 $/\$
40	39	ad4ant13 1292	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
41	40	eqcomd 2628	. . . . . . . . . . 11 $/\$ $/\$ $/\$
42		fveq2 6191	. . . . . . . . . . . 12
43		fveq2 6191	. . . . . . . . . . . 12
44	42, 43	eqeq12d 2637	. . . . . . . . . . 11
45	41, 44	syl5ibrcom 237	. . . . . . . . . 10 $/\$ $/\$ $/\$
46		fntpb.c	. . . . . . . . . . . . . 14
47	46, 24	fvtp3 6462	. . . . . . . . . . . . 13 $/\$
48	47	ad4ant23 1297	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
49	48	eqcomd 2628	. . . . . . . . . . 11 $/\$ $/\$ $/\$
50		fveq2 6191	. . . . . . . . . . . 12
51		fveq2 6191	. . . . . . . . . . . 12
52	50, 51	eqeq12d 2637	. . . . . . . . . . 11
53	49, 52	syl5ibrcom 237	. . . . . . . . . 10 $/\$ $/\$ $/\$
54	38, 45, 53	3jaod 1392	. . . . . . . . 9 $/\$ $/\$ $/\$ $\/$ $\/$
55	31, 54	syl5bi 232	. . . . . . . 8 $/\$ $/\$ $/\$
56	29, 55	sylbid 230	. . . . . . 7 $/\$ $/\$ $/\$
57	56	ralrimiv 2965	. . . . . 6 $/\$ $/\$ $/\$
58		fnfun 5988	. . . . . . 7
59	1, 2, 46, 22, 23, 24	funtp 5945	. . . . . . . 8 $/\$ $/\$
60	59	3expa 1265	. . . . . . 7 $/\$ $/\$
61		eqfunfv 6316	. . . . . . 7 $/\$ $/\$
62	58, 60, 61	syl2anr 495	. . . . . 6 $/\$ $/\$ $/\$ $/\$
63	27, 57, 62	mpbir2and 957	. . . . 5 $/\$ $/\$ $/\$
64	1, 2, 46, 22, 23, 24	fntp 5949	. . . . . . 7 $/\$ $/\$
65	64	3expa 1265	. . . . . 6 $/\$ $/\$
66		fneq1 5979	. . . . . . 7
67	66	biimprd 238	. . . . . 6
68	65, 67	mpan9 486	. . . . 5 $/\$ $/\$ $/\$
69	63, 68	impbida 877	. . . 4 $/\$ $/\$
70	69	expcom 451	. . 3 $/\$
71	20, 70	pm2.61ine 2877	. 2 $/\$
72	1, 46	fnprb 6472	. . . . 5
73		tpidm12 4290	. . . . . . 7
74	73	eqcomi 2631	. . . . . 6
75	74	fneq2i 5986	. . . . 5
76		tpidm12 4290	. . . . . . 7
77	76	eqcomi 2631	. . . . . 6
78	77	eqeq2i 2634	. . . . 5
79	72, 75, 78	3bitr3i 290	. . . 4
80	79	a1i 11	. . 3
81		tpeq2 4278	. . . 4
82	81	fneq2d 5982	. . 3
83		id 22	. . . . . 6
84		fveq2 6191	. . . . . 6
85	83, 84	opeq12d 4410	. . . . 5
86	85	tpeq2d 4281	. . . 4
87	86	eqeq2d 2632	. . 3
88	80, 82, 87	3bitr3d 298	. 2
89		tpidm13 4291	. . . . . . 7
90	89	eqcomi 2631	. . . . . 6
91	90	fneq2i 5986	. . . . 5
92		tpidm13 4291	. . . . . . 7
93	92	eqcomi 2631	. . . . . 6
94	93	eqeq2i 2634	. . . . 5
95	3, 91, 94	3bitr3i 290	. . . 4
96	95	a1i 11	. . 3
97		tpeq3 4279	. . . 4
98	97	fneq2d 5982	. . 3
99		id 22	. . . . . 6
100		fveq2 6191	. . . . . 6
101	99, 100	opeq12d 4410	. . . . 5
102	101	tpeq3d 4282	. . . 4
103	102	eqeq2d 2632	. . 3
104	96, 98, 103	3bitr3d 298	. 2
105	71, 88, 104	pm2.61iine 2884	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990

wne 2794

wral 2912

cvv 3200

cpr 4179

ctp 4181

cop 4183

cdm 5114

wfun 5882

wfn 5883

cfv 5888

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

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-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-sn 4178 df-pr 4180 df-tp 4182 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-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

This theorem is referenced by: wrd3tpop 13691

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