Theorem 2ffzoeq 32605

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 2461	. . . . . . . . . . . 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 10896	. . . . . . . . . . . . . . . . . . 19 |- 0 e. ZZ
8		nn0z 10908	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> N e. ZZ )
9	8	adantl 466	. . . . . . . . . . . . . . . . . . 19 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. ZZ )
10		fzon 11845	. . . . . . . . . . . . . . . . . . 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 10842	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> 0 <_ N )
13		0red 9614	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 0 e. RR )
14		nn0re 10825	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. RR )
15	13, 14	letri3d 9744	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> ( 0 = N <-> ( 0 <_ N /\ N <_ 0 ) ) )
16	15	biimprd 223	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> ( ( 0 <_ N /\ N <_ 0 ) -> 0 = N ) )
17	12, 16	mpand 675	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN0 -> ( N <_ 0 -> 0 = N ) )
18	17	adantl 466	. . . . . . . . . . . . . . . . . 18 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N <_ 0 -> 0 = N ) )
19	11, 18	sylbird 235	. . . . . . . . . . . . . . . . 17 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 0..^ N ) = (/) -> 0 = N ) )
20	6, 19	syl5com 30	. . . . . . . . . . . . . . . 16 |- ( ( P Fn ( 0..^ N ) /\ P Fn (/) ) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) )
21	20	ex 434	. . . . . . . . . . . . . . 15 |- ( P Fn ( 0..^ N ) -> ( P Fn (/) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
22	5, 21	syl 16	. . . . . . . . . . . . . 14 |- ( P : ( 0..^ N ) --> Y -> ( P Fn (/) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
23	4, 22	syl5com 30	. . . . . . . . . . . . 13 |- ( P : (/) --> Y -> ( P : ( 0..^ N ) --> Y -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
24	3, 23	sylbir 213	. . . . . . . . . . . 12 |- ( P = (/) -> ( P : ( 0..^ N ) --> Y -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
25	24	imp 429	. . . . . . . . . . 11 |- ( ( P = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) )
26	2, 25	syl6bi 228	. . . . . . . . . 10 |- ( F = P -> ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
27	26	com3l 81	. . . . . . . . 9 |- ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( F = P -> 0 = N ) ) )
28	27	a1i 11	. . . . . . . 8 |- ( M = 0 -> ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( F = P -> 0 = N ) ) ) )
29		oveq2 6304	. . . . . . . . . . . 12 |- ( M = 0 -> ( 0..^ M ) = ( 0..^ 0 ) )
30		fzo0 11848	. . . . . . . . . . . 12 |- ( 0..^ 0 ) = (/)
31	29, 30	syl6eq 2514	. . . . . . . . . . 11 |- ( M = 0 -> ( 0..^ M ) = (/) )
32	31	feq2d 5724	. . . . . . . . . 10 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : (/) --> X ) )
33		f0bi 5774	. . . . . . . . . 10 |- ( F : (/) --> X <-> F = (/) )
34	32, 33	syl6bb 261	. . . . . . . . 9 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F = (/) ) )
35	34	anbi1d 704	. . . . . . . 8 |- ( M = 0 -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) <-> ( F = (/) /\ P : ( 0..^ N ) --> Y ) ) )
36		eqeq1 2461	. . . . . . . . . 10 |- ( M = 0 -> ( M = N <-> 0 = N ) )
37	36	imbi2d 316	. . . . . . . . 9 |- ( M = 0 -> ( ( F = P -> M = N ) <-> ( F = P -> 0 = N ) ) )
38	37	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 ) ) ) )
39	28, 35, 38	3imtr4d 268	. . . . . . 7 |- ( M = 0 -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( F = P -> M = N ) ) ) )
40	39	com3l 81	. . . . . 6 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( M = 0 -> ( F = P -> M = N ) ) ) )
41	40	impcom 430	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( M = 0 -> ( F = P -> M = N ) ) )
42	41	impcom 430	. . . 4 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F = P -> M = N ) )
43	29	feq2d 5724	. . . . . . . . . . . 12 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : ( 0..^ 0 ) --> X ) )
44	30	feq2i 5730	. . . . . . . . . . . . 13 |- ( F : ( 0..^ 0 ) --> X <-> F : (/) --> X )
45	44, 33	bitri 249	. . . . . . . . . . . 12 |- ( F : ( 0..^ 0 ) --> X <-> F = (/) )
46	43, 45	syl6bb 261	. . . . . . . . . . 11 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F = (/) ) )
47	46	adantr 465	. . . . . . . . . 10 |- ( ( M = 0 /\ M = N ) -> ( F : ( 0..^ M ) --> X <-> F = (/) ) )
48		eqeq1 2461	. . . . . . . . . . . 12 |- ( M = N -> ( M = 0 <-> N = 0 ) )
49	48	biimpac 486	. . . . . . . . . . 11 |- ( ( M = 0 /\ M = N ) -> N = 0 )
50		oveq2 6304	. . . . . . . . . . . . 13 |- ( N = 0 -> ( 0..^ N ) = ( 0..^ 0 ) )
51	50	feq2d 5724	. . . . . . . . . . . 12 |- ( N = 0 -> ( P : ( 0..^ N ) --> Y <-> P : ( 0..^ 0 ) --> Y ) )
52	30	feq2i 5730	. . . . . . . . . . . . 13 |- ( P : ( 0..^ 0 ) --> Y <-> P : (/) --> Y )
53	52, 3	bitri 249	. . . . . . . . . . . 12 |- ( P : ( 0..^ 0 ) --> Y <-> P = (/) )
54	51, 53	syl6bb 261	. . . . . . . . . . 11 |- ( N = 0 -> ( P : ( 0..^ N ) --> Y <-> P = (/) ) )
55	49, 54	syl 16	. . . . . . . . . 10 |- ( ( M = 0 /\ M = N ) -> ( P : ( 0..^ N ) --> Y <-> P = (/) ) )
56	47, 55	anbi12d 710	. . . . . . . . 9 |- ( ( M = 0 /\ M = N ) -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) <-> ( F = (/) /\ P = (/) ) ) )
57		eqtr3 2485	. . . . . . . . 9 |- ( ( F = (/) /\ P = (/) ) -> F = P )
58	56, 57	syl6bi 228	. . . . . . . 8 |- ( ( M = 0 /\ M = N ) -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> F = P ) )
59	58	com12 31	. . . . . . 7 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( ( M = 0 /\ M = N ) -> F = P ) )
60	59	expd 436	. . . . . 6 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( M = 0 -> ( M = N -> F = P ) ) )
61	60	adantl 466	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( M = 0 -> ( M = N -> F = P ) ) )
62	61	impcom 430	. . . 4 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( M = N -> F = P ) )
63	42, 62	impbid 191	. . 3 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F = P <-> M = N ) )
64		ral0 3937	. . . . . 6 |- A. i e. (/) ( F ` i ) = ( P ` i )
65	31	raleqdv 3060	. . . . . 6 |- ( M = 0 -> ( A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) <-> A. i e. (/) ( F ` i ) = ( P ` i ) ) )
66	64, 65	mpbiri 233	. . . . 5 |- ( M = 0 -> A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) )
67	66	biantrud 507	. . . 4 |- ( M = 0 -> ( M = N <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
68	67	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 ) ) ) )
69	63, 68	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 ) ) ) )
70		ffn 5737	. . . . . . 7 |- ( F : ( 0..^ M ) --> X -> F Fn ( 0..^ M ) )
71	70, 5	anim12i 566	. . . . . 6 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( F Fn ( 0..^ M ) /\ P Fn ( 0..^ N ) ) )
72	71	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 ) ) )
73	72	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 ) ) )
74		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 ) ) ) )
75	73, 74	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 ) ) ) )
76		df-ne 2654	. . . . . 6 |- ( M =/= 0 <-> -. M = 0 )
77		elnnne0 10830	. . . . . . . 8 |- ( M e. NN <-> ( M e. NN0 /\ M =/= 0 ) )
78		0zd 10897	. . . . . . . . . . . . . . 15 |- ( M e. NN -> 0 e. ZZ )
79		nnz 10907	. . . . . . . . . . . . . . 15 |- ( M e. NN -> M e. ZZ )
80		nngt0 10585	. . . . . . . . . . . . . . 15 |- ( M e. NN -> 0 < M )
81	78, 79, 80	3jca 1176	. . . . . . . . . . . . . 14 |- ( M e. NN -> ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) )
82	81	adantr 465	. . . . . . . . . . . . 13 |- ( ( M e. NN /\ N e. NN0 ) -> ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) )
83		fzoopth 32604	. . . . . . . . . . . . 13 |- ( ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) -> ( ( 0..^ M ) = ( 0..^ N ) <-> ( 0 = 0 /\ M = N ) ) )
84	82, 83	syl 16	. . . . . . . . . . . 12 |- ( ( M e. NN /\ N e. NN0 ) -> ( ( 0..^ M ) = ( 0..^ N ) <-> ( 0 = 0 /\ M = N ) ) )
85		simpr 461	. . . . . . . . . . . 12 |- ( ( 0 = 0 /\ M = N ) -> M = N )
86	84, 85	syl6bi 228	. . . . . . . . . . 11 |- ( ( M e. NN /\ N e. NN0 ) -> ( ( 0..^ M ) = ( 0..^ N ) -> M = N ) )
87	86	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 ) ) ) )
88		oveq2 6304	. . . . . . . . . . 11 |- ( M = N -> ( 0..^ M ) = ( 0..^ N ) )
89	88	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 ) ) )
90	87, 89	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 ) ) ) )
91	90	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 ) ) ) ) )
92	77, 91	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 ) ) ) ) )
93	92	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 ) ) ) ) )
94	76, 93	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 ) ) ) ) )
95	94	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 ) ) ) ) )
96	95	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 ) ) ) )
97	75, 96	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 ) ) ) )
98	69, 97	pm2.61ian 790	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 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   (/)c0 3793   class class class wbr 4456   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   < clt 9645   <_ cle 9646   NNcn 10556   NN0cn0 10816   ZZcz 10885  ..^cfzo 11821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822

This theorem is referenced by: (None)