Theorem 2ffzoeq 30382

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 2458	. . . . . . . . . . . 12 |- ( F = P -> ( F = (/) <-> P = (/) ) )
2	1	anbi1d 704	. . . . . . . . . . 11 |- ( F = P -> ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) <-> ( P = (/) /\ P : ( 0..^ N ) --> Y ) ) )
3		f0bi 5705	. . . . . . . . . . . . 13 |- ( P : (/) --> Y <-> P = (/) )
4		ffn 5670	. . . . . . . . . . . . . 14 |- ( P : (/) --> Y -> P Fn (/) )
5		ffn 5670	. . . . . . . . . . . . . . 15 |- ( P : ( 0..^ N ) --> Y -> P Fn ( 0..^ N ) )
6		fndmu 5623	. . . . . . . . . . . . . . . . 17 |- ( ( P Fn ( 0..^ N ) /\ P Fn (/) ) -> ( 0..^ N ) = (/) )
7		0z 10771	. . . . . . . . . . . . . . . . . . 19 |- 0 e. ZZ
8		nn0z 10783	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> N e. ZZ )
9	8	adantl 466	. . . . . . . . . . . . . . . . . . 19 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. ZZ )
10		fzon 11691	. . . . . . . . . . . . . . . . . . 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 10719	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> 0 <_ N )
13		0re 9500	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 e. RR
14	13	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 0 e. RR )
15		nn0re 10702	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. RR )
16	14, 15	letri3d 9630	. . . . . . . . . . . . . . . . . . . . 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 6211	. . . . . . . . . . . 12 |- ( M = 0 -> ( 0..^ M ) = ( 0..^ 0 ) )
31		fzo0 11693	. . . . . . . . . . . 12 |- ( 0..^ 0 ) = (/)
32	30, 31	syl6eq 2511	. . . . . . . . . . 11 |- ( M = 0 -> ( 0..^ M ) = (/) )
33	32	feq2d 5658	. . . . . . . . . 10 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : (/) --> X ) )
34		f0bi 5705	. . . . . . . . . 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 2458	. . . . . . . . . 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 5658	. . . . . . . . . . . 12 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : ( 0..^ 0 ) --> X ) )
45	31	feq2i 5663	. . . . . . . . . . . . 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 2458	. . . . . . . . . . . . 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 6211	. . . . . . . . . . . . 13 |- ( N = 0 -> ( 0..^ N ) = ( 0..^ 0 ) )
53	52	feq2d 5658	. . . . . . . . . . . 12 |- ( N = 0 -> ( P : ( 0..^ N ) --> Y <-> P : ( 0..^ 0 ) --> Y ) )
54	31	feq2i 5663	. . . . . . . . . . . . 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 2482	. . . . . . . . 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 3895	. . . . . 6 |- A. i e. (/) ( F ` i ) = ( P ` i )
67	32	raleqdv 3029	. . . . . 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 5670	. . . . . . 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 5910	. . . 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 2650	. . . . . 6 |- ( M =/= 0 <-> -. M = 0 )
79		elnnne0 10707	. . . . . . . 8 |- ( M e. NN <-> ( M e. NN0 /\ M =/= 0 ) )
80		0zd 10772	. . . . . . . . . . . . . . 15 |- ( M e. NN -> 0 e. ZZ )
81		nnz 10782	. . . . . . . . . . . . . . 15 |- ( M e. NN -> M e. ZZ )
82		nngt0 10465	. . . . . . . . . . . . . . 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 30381	. . . . . . . . . . . . 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 6211	. . . . . . . . . . 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 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   (/)c0 3748   class class class wbr 4403   Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   RRcr 9395   0cc0 9396   < clt 9532   <_ cle 9533   NNcn 10436   NN0cn0 10693   ZZcz 10760  ..^cfzo 11668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669

This theorem is referenced by: (None)