Theorem 2ffzoeq 32122

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 2471	. . . . . . . . . . . 12 |- ( F = P -> ( F = (/) <-> P = (/) ) )
2	1	anbi1d 704	. . . . . . . . . . 11 |- ( F = P -> ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) <-> ( P = (/) /\ P : ( 0..^ N ) --> Y ) ) )
3		f0bi 5774	. . . . . . . . . . . . 13 |- ( P : (/) --> Y <-> P = (/) )
4		ffn 5737	. . . . . . . . . . . . . 14 |- ( P : (/) --> Y -> P Fn (/) )
5		ffn 5737	. . . . . . . . . . . . . . 15 |- ( P : ( 0..^ N ) --> Y -> P Fn ( 0..^ N ) )
6		fndmu 5688	. . . . . . . . . . . . . . . . 17 |- ( ( P Fn ( 0..^ N ) /\ P Fn (/) ) -> ( 0..^ N ) = (/) )
7		0z 10887	. . . . . . . . . . . . . . . . . . 19 |- 0 e. ZZ
8		nn0z 10899	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> N e. ZZ )
9	8	adantl 466	. . . . . . . . . . . . . . . . . . 19 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. ZZ )
10		fzon 11827	. . . . . . . . . . . . . . . . . . 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 10833	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> 0 <_ N )
13		0re 9608	. . . . . . . . . . . . . . . . . . . . . . 23 |- 0 e. RR
14	13	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 0 e. RR )
15		nn0re 10816	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. RR )
16	14, 15	letri3d 9738	. . . . . . . . . . . . . . . . . . . . 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 6303	. . . . . . . . . . . 12 |- ( M = 0 -> ( 0..^ M ) = ( 0..^ 0 ) )
31		fzo0 11829	. . . . . . . . . . . 12 |- ( 0..^ 0 ) = (/)
32	30, 31	syl6eq 2524	. . . . . . . . . . 11 |- ( M = 0 -> ( 0..^ M ) = (/) )
33	32	feq2d 5724	. . . . . . . . . 10 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : (/) --> X ) )
34		f0bi 5774	. . . . . . . . . 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 2471	. . . . . . . . . 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 5724	. . . . . . . . . . . 12 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : ( 0..^ 0 ) --> X ) )
45	31	feq2i 5730	. . . . . . . . . . . . 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 2471	. . . . . . . . . . . . 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 6303	. . . . . . . . . . . . 13 |- ( N = 0 -> ( 0..^ N ) = ( 0..^ 0 ) )
53	52	feq2d 5724	. . . . . . . . . . . 12 |- ( N = 0 -> ( P : ( 0..^ N ) --> Y <-> P : ( 0..^ 0 ) --> Y ) )
54	31	feq2i 5730	. . . . . . . . . . . . 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 2495	. . . . . . . . 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 3938	. . . . . 6 |- A. i e. (/) ( F ` i ) = ( P ` i )
67	32	raleqdv 3069	. . . . . 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 5737	. . . . . . 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 5983	. . . 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 2664	. . . . . 6 |- ( M =/= 0 <-> -. M = 0 )
79		elnnne0 10821	. . . . . . . 8 |- ( M e. NN <-> ( M e. NN0 /\ M =/= 0 ) )
80		0zd 10888	. . . . . . . . . . . . . . 15 |- ( M e. NN -> 0 e. ZZ )
81		nnz 10898	. . . . . . . . . . . . . . 15 |- ( M e. NN -> M e. ZZ )
82		nngt0 10577	. . . . . . . . . . . . . . 15 |- ( M e. NN -> 0 < M )
83	80, 81, 82	3jca 1176	. . . . . . . . . . . . . 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 32121	. . . . . . . . . . . . 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 6303	. . . . . . . . . . 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 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2817   (/)c0 3790   class class class wbr 4453   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   < clt 9640   <_ cle 9641   NNcn 10548   NN0cn0 10807   ZZcz 10876  ..^cfzo 11804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805

This theorem is referenced by: (None)