Theorem 2ffzoeq 30167

Description: Two functions over a half-open range of nonnegative integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)

Assertion

Ref	Expression
2ffzoeq	|- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( F = P <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )

Distinct variable groups:   i, F   i, M   P, i

Allowed substitution hints:   N( i)   X( i)   Y( i)

Proof of Theorem 2ffzoeq

Step	Hyp	Ref	Expression
1		eqeq1 2444	. . . . . . . . . . . 12 |- ( F = P -> ( F = (/) <-> P = (/) ) )
2	1	anbi1d 704	. . . . . . . . . . 11 |- ( F = P -> ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) <-> ( P = (/) /\ P : ( 0..^ N ) --> Y ) ) )
3		f0bi 5589	. . . . . . . . . . . . 13 |- ( P : (/) --> Y <-> P = (/) )
4		ffn 5554	. . . . . . . . . . . . . 14 |- ( P : (/) --> Y -> P Fn (/) )
5		ffn 5554	. . . . . . . . . . . . . . 15 |- ( P : ( 0..^ N ) --> Y -> P Fn ( 0..^ N ) )
6		fndmu 5507	. . . . . . . . . . . . . . . . 17 |- ( ( P Fn ( 0..^ N ) /\ P Fn (/) ) -> ( 0..^ N ) = (/) )
7		0z 10649	. . . . . . . . . . . . . . . . . . 19 |- 0 e. ZZ
8		nn0z 10661	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> N e. ZZ )
9	8	adantl 466	. . . . . . . . . . . . . . . . . . 19 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. ZZ )
10		fzon 11563	. . . . . . . . . . . . . . . . . . 19 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( N <_ 0 <-> ( 0..^ N ) = (/) ) )
11	7, 9, 10	sylancr 663	. . . . . . . . . . . . . . . . . 18 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N <_ 0 <-> ( 0..^ N ) = (/) ) )
12		nn0ge0 10597	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> 0 <_ N )
13		0re 9378	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 e. RR
14	13	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 0 e. RR )
15		nn0re 10580	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. RR )
16	14, 15	letri3d 9508	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> ( 0 = N <-> ( 0 <_ N /\ N <_ 0 ) ) )
17	16	biimprd 223	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> ( ( 0 <_ N /\ N <_ 0 ) -> 0 = N ) )
18	12, 17	mpand 675	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN0 -> ( N <_ 0 -> 0 = N ) )
19	18	adantl 466	. . . . . . . . . . . . . . . . . 18 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N <_ 0 -> 0 = N ) )
20	11, 19	sylbird 235	. . . . . . . . . . . . . . . . 17 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 0..^ N ) = (/) -> 0 = N ) )
21	6, 20	syl5com 30	. . . . . . . . . . . . . . . 16 |- ( ( P Fn ( 0..^ N ) /\ P Fn (/) ) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) )
22	21	ex 434	. . . . . . . . . . . . . . 15 |- ( P Fn ( 0..^ N ) -> ( P Fn (/) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
23	5, 22	syl 16	. . . . . . . . . . . . . 14 |- ( P : ( 0..^ N ) --> Y -> ( P Fn (/) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
24	4, 23	syl5com 30	. . . . . . . . . . . . 13 |- ( P : (/) --> Y -> ( P : ( 0..^ N ) --> Y -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
25	3, 24	sylbir 213	. . . . . . . . . . . 12 |- ( P = (/) -> ( P : ( 0..^ N ) --> Y -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
26	25	imp 429	. . . . . . . . . . 11 |- ( ( P = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) )
27	2, 26	syl6bi 228	. . . . . . . . . 10 |- ( F = P -> ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
28	27	com3l 81	. . . . . . . . 9 |- ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( F = P -> 0 = N ) ) )
29	28	a1i 11	. . . . . . . 8 |- ( M = 0 -> ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( F = P -> 0 = N ) ) ) )
30		oveq2 6094	. . . . . . . . . . . 12 |- ( M = 0 -> ( 0..^ M ) = ( 0..^ 0 ) )
31		fzo0 11565	. . . . . . . . . . . 12 |- ( 0..^ 0 ) = (/)
32	30, 31	syl6eq 2486	. . . . . . . . . . 11 |- ( M = 0 -> ( 0..^ M ) = (/) )
33	32	feq2d 5542	. . . . . . . . . 10 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : (/) --> X ) )
34		f0bi 5589	. . . . . . . . . 10 |- ( F : (/) --> X <-> F = (/) )
35	33, 34	syl6bb 261	. . . . . . . . 9 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F = (/) ) )
36	35	anbi1d 704	. . . . . . . 8 |- ( M = 0 -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) <-> ( F = (/) /\ P : ( 0..^ N ) --> Y ) ) )
37		eqeq1 2444	. . . . . . . . . 10 |- ( M = 0 -> ( M = N <-> 0 = N ) )
38	37	imbi2d 316	. . . . . . . . 9 |- ( M = 0 -> ( ( F = P -> M = N ) <-> ( F = P -> 0 = N ) ) )
39	38	imbi2d 316	. . . . . . . 8 |- ( M = 0 -> ( ( ( M e. NN0 /\ N e. NN0 ) -> ( F = P -> M = N ) ) <-> ( ( M e. NN0 /\ N e. NN0 ) -> ( F = P -> 0 = N ) ) ) )
40	29, 36, 39	3imtr4d 268	. . . . . . 7 |- ( M = 0 -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( F = P -> M = N ) ) ) )
41	40	com3l 81	. . . . . 6 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( M = 0 -> ( F = P -> M = N ) ) ) )
42	41	impcom 430	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( M = 0 -> ( F = P -> M = N ) ) )
43	42	impcom 430	. . . 4 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F = P -> M = N ) )
44	30	feq2d 5542	. . . . . . . . . . . 12 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : ( 0..^ 0 ) --> X ) )
45	31	feq2i 5547	. . . . . . . . . . . . 13 |- ( F : ( 0..^ 0 ) --> X <-> F : (/) --> X )
46	45, 34	bitri 249	. . . . . . . . . . . 12 |- ( F : ( 0..^ 0 ) --> X <-> F = (/) )
47	44, 46	syl6bb 261	. . . . . . . . . . 11 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F = (/) ) )
48	47	adantr 465	. . . . . . . . . 10 |- ( ( M = 0 /\ M = N ) -> ( F : ( 0..^ M ) --> X <-> F = (/) ) )
49		eqeq1 2444	. . . . . . . . . . . . 13 |- ( M = N -> ( M = 0 <-> N = 0 ) )
50	49	biimpd 207	. . . . . . . . . . . 12 |- ( M = N -> ( M = 0 -> N = 0 ) )
51	50	impcom 430	. . . . . . . . . . 11 |- ( ( M = 0 /\ M = N ) -> N = 0 )
52		oveq2 6094	. . . . . . . . . . . . 13 |- ( N = 0 -> ( 0..^ N ) = ( 0..^ 0 ) )
53	52	feq2d 5542	. . . . . . . . . . . 12 |- ( N = 0 -> ( P : ( 0..^ N ) --> Y <-> P : ( 0..^ 0 ) --> Y ) )
54	31	feq2i 5547	. . . . . . . . . . . . 13 |- ( P : ( 0..^ 0 ) --> Y <-> P : (/) --> Y )
55	54, 3	bitri 249	. . . . . . . . . . . 12 |- ( P : ( 0..^ 0 ) --> Y <-> P = (/) )
56	53, 55	syl6bb 261	. . . . . . . . . . 11 |- ( N = 0 -> ( P : ( 0..^ N ) --> Y <-> P = (/) ) )
57	51, 56	syl 16	. . . . . . . . . 10 |- ( ( M = 0 /\ M = N ) -> ( P : ( 0..^ N ) --> Y <-> P = (/) ) )
58	48, 57	anbi12d 710	. . . . . . . . 9 |- ( ( M = 0 /\ M = N ) -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) <-> ( F = (/) /\ P = (/) ) ) )
59		eqtr3 2457	. . . . . . . . 9 |- ( ( F = (/) /\ P = (/) ) -> F = P )
60	58, 59	syl6bi 228	. . . . . . . 8 |- ( ( M = 0 /\ M = N ) -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> F = P ) )
61	60	com12 31	. . . . . . 7 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( ( M = 0 /\ M = N ) -> F = P ) )
62	61	expd 436	. . . . . 6 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( M = 0 -> ( M = N -> F = P ) ) )
63	62	adantl 466	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( M = 0 -> ( M = N -> F = P ) ) )
64	63	impcom 430	. . . 4 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( M = N -> F = P ) )
65	43, 64	impbid 191	. . 3 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F = P <-> M = N ) )
66		ral0 3779	. . . . . 6 |- A. i e. (/) ( F ` i ) = ( P ` i )
67	32	raleqdv 2918	. . . . . 6 |- ( M = 0 -> ( A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) <-> A. i e. (/) ( F ` i ) = ( P ` i ) ) )
68	66, 67	mpbiri 233	. . . . 5 |- ( M = 0 -> A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) )
69	68	biantrud 507	. . . 4 |- ( M = 0 -> ( M = N <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
70	69	adantr 465	. . 3 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( M = N <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
71	65, 70	bitrd 253	. 2 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F = P <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
72		ffn 5554	. . . . . . 7 |- ( F : ( 0..^ M ) --> X -> F Fn ( 0..^ M ) )
73	72, 5	anim12i 566	. . . . . 6 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( F Fn ( 0..^ M ) /\ P Fn ( 0..^ N ) ) )
74	73	adantl 466	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( F Fn ( 0..^ M ) /\ P Fn ( 0..^ N ) ) )
75	74	adantl 466	. . . 4 |- ( ( -. M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F Fn ( 0..^ M ) /\ P Fn ( 0..^ N ) ) )
76		eqfnfv2 5793	. . . 4 |- ( ( F Fn ( 0..^ M ) /\ P Fn ( 0..^ N ) ) -> ( F = P <-> ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
77	75, 76	syl 16	. . 3 |- ( ( -. M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F = P <-> ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
78		df-ne 2603	. . . . . 6 |- ( M =/= 0 <-> -. M = 0 )
79		elnnne0 10585	. . . . . . . 8 |- ( M e. NN <-> ( M e. NN0 /\ M =/= 0 ) )
80		0zd 10650	. . . . . . . . . . . . . . 15 |- ( M e. NN -> 0 e. ZZ )
81		nnz 10660	. . . . . . . . . . . . . . 15 |- ( M e. NN -> M e. ZZ )
82		nngt0 10343	. . . . . . . . . . . . . . 15 |- ( M e. NN -> 0 < M )
83	80, 81, 82	3jca 1168	. . . . . . . . . . . . . 14 |- ( M e. NN -> ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) )
84	83	adantr 465	. . . . . . . . . . . . 13 |- ( ( M e. NN /\ N e. NN0 ) -> ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) )
85		fzoopth 30166	. . . . . . . . . . . . 13 |- ( ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) -> ( ( 0..^ M ) = ( 0..^ N ) <-> ( 0 = 0 /\ M = N ) ) )
86	84, 85	syl 16	. . . . . . . . . . . 12 |- ( ( M e. NN /\ N e. NN0 ) -> ( ( 0..^ M ) = ( 0..^ N ) <-> ( 0 = 0 /\ M = N ) ) )
87		simpr 461	. . . . . . . . . . . 12 |- ( ( 0 = 0 /\ M = N ) -> M = N )
88	86, 87	syl6bi 228	. . . . . . . . . . 11 |- ( ( M e. NN /\ N e. NN0 ) -> ( ( 0..^ M ) = ( 0..^ N ) -> M = N ) )
89	88	anim1d 564	. . . . . . . . . 10 |- ( ( M e. NN /\ N e. NN0 ) -> ( ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) -> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
90		oveq2 6094	. . . . . . . . . . 11 |- ( M = N -> ( 0..^ M ) = ( 0..^ N ) )
91	90	anim1i 568	. . . . . . . . . 10 |- ( ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) -> ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) )
92	89, 91	impbid1 203	. . . . . . . . 9 |- ( ( M e. NN /\ N e. NN0 ) -> ( ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
93	92	ex 434	. . . . . . . 8 |- ( M e. NN -> ( N e. NN0 -> ( ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) ) )
94	79, 93	sylbir 213	. . . . . . 7 |- ( ( M e. NN0 /\ M =/= 0 ) -> ( N e. NN0 -> ( ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) ) )
95	94	impancom 440	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M =/= 0 -> ( ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) ) )
96	78, 95	syl5bir 218	. . . . 5 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( -. M = 0 -> ( ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) ) )
97	96	adantr 465	. . . 4 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( -. M = 0 -> ( ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) ) )
98	97	impcom 430	. . 3 |- ( ( -. M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( ( ( 0..^ M ) = ( 0..^ N ) /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
99	77, 98	bitrd 253	. 2 |- ( ( -. M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F = P <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
100	71, 99	pm2.61ian 788	1 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( F = P <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2601   A.wral 2710   (/)c0 3632   class class class wbr 4287   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   RRcr 9273   0cc0 9274   < clt 9410   <_ cle 9411   NNcn 10314   NN0cn0 10571   ZZcz 10638  ..^cfzo 11540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541

This theorem is referenced by: (None)