Theorem 2ffzoeq 38752

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 2426	. . . . . . . . . . . 12 |- ( F = P -> ( F = (/) <-> P = (/) ) )
2	1	anbi1d 709	. . . . . . . . . . 11 |- ( F = P -> ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) <-> ( P = (/) /\ P : ( 0..^ N ) --> Y ) ) )
3		f0bi 5779	. . . . . . . . . . . . 13 |- ( P : (/) --> Y <-> P = (/) )
4		ffn 5742	. . . . . . . . . . . . . 14 |- ( P : (/) --> Y -> P Fn (/) )
5		ffn 5742	. . . . . . . . . . . . . . 15 |- ( P : ( 0..^ N ) --> Y -> P Fn ( 0..^ N ) )
6		fndmu 5691	. . . . . . . . . . . . . . . . 17 |- ( ( P Fn ( 0..^ N ) /\ P Fn (/) ) -> ( 0..^ N ) = (/) )
7		0z 10948	. . . . . . . . . . . . . . . . . . 19 |- 0 e. ZZ
8		nn0z 10960	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> N e. ZZ )
9	8	adantl 467	. . . . . . . . . . . . . . . . . . 19 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. ZZ )
10		fzon 11939	. . . . . . . . . . . . . . . . . . 19 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( N <_ 0 <-> ( 0..^ N ) = (/) ) )
11	7, 9, 10	sylancr 667	. . . . . . . . . . . . . . . . . 18 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N <_ 0 <-> ( 0..^ N ) = (/) ) )
12		nn0ge0 10895	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> 0 <_ N )
13		0red 9644	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> 0 e. RR )
14		nn0re 10878	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN0 -> N e. RR )
15	13, 14	letri3d 9777	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> ( 0 = N <-> ( 0 <_ N /\ N <_ 0 ) ) )
16	15	biimprd 226	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> ( ( 0 <_ N /\ N <_ 0 ) -> 0 = N ) )
17	12, 16	mpand 679	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN0 -> ( N <_ 0 -> 0 = N ) )
18	17	adantl 467	. . . . . . . . . . . . . . . . . 18 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( N <_ 0 -> 0 = N ) )
19	11, 18	sylbird 238	. . . . . . . . . . . . . . . . 17 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( ( 0..^ N ) = (/) -> 0 = N ) )
20	6, 19	syl5com 31	. . . . . . . . . . . . . . . 16 |- ( ( P Fn ( 0..^ N ) /\ P Fn (/) ) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) )
21	20	ex 435	. . . . . . . . . . . . . . 15 |- ( P Fn ( 0..^ N ) -> ( P Fn (/) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
22	5, 21	syl 17	. . . . . . . . . . . . . 14 |- ( P : ( 0..^ N ) --> Y -> ( P Fn (/) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
23	4, 22	syl5com 31	. . . . . . . . . . . . 13 |- ( P : (/) --> Y -> ( P : ( 0..^ N ) --> Y -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
24	3, 23	sylbir 216	. . . . . . . . . . . 12 |- ( P = (/) -> ( P : ( 0..^ N ) --> Y -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
25	24	imp 430	. . . . . . . . . . 11 |- ( ( P = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) )
26	2, 25	syl6bi 231	. . . . . . . . . 10 |- ( F = P -> ( ( F = (/) /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> 0 = N ) ) )
27	26	com3l 84	. . . . . . . . 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 6309	. . . . . . . . . . . 12 |- ( M = 0 -> ( 0..^ M ) = ( 0..^ 0 ) )
30		fzo0 11942	. . . . . . . . . . . 12 |- ( 0..^ 0 ) = (/)
31	29, 30	syl6eq 2479	. . . . . . . . . . 11 |- ( M = 0 -> ( 0..^ M ) = (/) )
32	31	feq2d 5729	. . . . . . . . . 10 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : (/) --> X ) )
33		f0bi 5779	. . . . . . . . . 10 |- ( F : (/) --> X <-> F = (/) )
34	32, 33	syl6bb 264	. . . . . . . . 9 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F = (/) ) )
35	34	anbi1d 709	. . . . . . . 8 |- ( M = 0 -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) <-> ( F = (/) /\ P : ( 0..^ N ) --> Y ) ) )
36		eqeq1 2426	. . . . . . . . . 10 |- ( M = 0 -> ( M = N <-> 0 = N ) )
37	36	imbi2d 317	. . . . . . . . 9 |- ( M = 0 -> ( ( F = P -> M = N ) <-> ( F = P -> 0 = N ) ) )
38	37	imbi2d 317	. . . . . . . 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 271	. . . . . . 7 |- ( M = 0 -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( F = P -> M = N ) ) ) )
40	39	com3l 84	. . . . . 6 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( ( M e. NN0 /\ N e. NN0 ) -> ( M = 0 -> ( F = P -> M = N ) ) ) )
41	40	impcom 431	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( M = 0 -> ( F = P -> M = N ) ) )
42	41	impcom 431	. . . 4 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F = P -> M = N ) )
43	29	feq2d 5729	. . . . . . . . . . . 12 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F : ( 0..^ 0 ) --> X ) )
44	30	feq2i 5735	. . . . . . . . . . . . 13 |- ( F : ( 0..^ 0 ) --> X <-> F : (/) --> X )
45	44, 33	bitri 252	. . . . . . . . . . . 12 |- ( F : ( 0..^ 0 ) --> X <-> F = (/) )
46	43, 45	syl6bb 264	. . . . . . . . . . 11 |- ( M = 0 -> ( F : ( 0..^ M ) --> X <-> F = (/) ) )
47	46	adantr 466	. . . . . . . . . 10 |- ( ( M = 0 /\ M = N ) -> ( F : ( 0..^ M ) --> X <-> F = (/) ) )
48		eqeq1 2426	. . . . . . . . . . . 12 |- ( M = N -> ( M = 0 <-> N = 0 ) )
49	48	biimpac 488	. . . . . . . . . . 11 |- ( ( M = 0 /\ M = N ) -> N = 0 )
50		oveq2 6309	. . . . . . . . . . . . 13 |- ( N = 0 -> ( 0..^ N ) = ( 0..^ 0 ) )
51	50	feq2d 5729	. . . . . . . . . . . 12 |- ( N = 0 -> ( P : ( 0..^ N ) --> Y <-> P : ( 0..^ 0 ) --> Y ) )
52	30	feq2i 5735	. . . . . . . . . . . . 13 |- ( P : ( 0..^ 0 ) --> Y <-> P : (/) --> Y )
53	52, 3	bitri 252	. . . . . . . . . . . 12 |- ( P : ( 0..^ 0 ) --> Y <-> P = (/) )
54	51, 53	syl6bb 264	. . . . . . . . . . 11 |- ( N = 0 -> ( P : ( 0..^ N ) --> Y <-> P = (/) ) )
55	49, 54	syl 17	. . . . . . . . . 10 |- ( ( M = 0 /\ M = N ) -> ( P : ( 0..^ N ) --> Y <-> P = (/) ) )
56	47, 55	anbi12d 715	. . . . . . . . 9 |- ( ( M = 0 /\ M = N ) -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) <-> ( F = (/) /\ P = (/) ) ) )
57		eqtr3 2450	. . . . . . . . 9 |- ( ( F = (/) /\ P = (/) ) -> F = P )
58	56, 57	syl6bi 231	. . . . . . . 8 |- ( ( M = 0 /\ M = N ) -> ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> F = P ) )
59	58	com12 32	. . . . . . 7 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( ( M = 0 /\ M = N ) -> F = P ) )
60	59	expd 437	. . . . . 6 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( M = 0 -> ( M = N -> F = P ) ) )
61	60	adantl 467	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) -> ( M = 0 -> ( M = N -> F = P ) ) )
62	61	impcom 431	. . . 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 193	. . 3 |- ( ( M = 0 /\ ( ( M e. NN0 /\ N e. NN0 ) /\ ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) ) ) -> ( F = P <-> M = N ) )
64		ral0 3902	. . . . . 6 |- A. i e. (/) ( F ` i ) = ( P ` i )
65	31	raleqdv 3031	. . . . . 6 |- ( M = 0 -> ( A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) <-> A. i e. (/) ( F ` i ) = ( P ` i ) ) )
66	64, 65	mpbiri 236	. . . . 5 |- ( M = 0 -> A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) )
67	66	biantrud 509	. . . 4 |- ( M = 0 -> ( M = N <-> ( M = N /\ A. i e. ( 0..^ M ) ( F ` i ) = ( P ` i ) ) ) )
68	67	adantr 466	. . 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 256	. 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 5742	. . . . . . 7 |- ( F : ( 0..^ M ) --> X -> F Fn ( 0..^ M ) )
71	70, 5	anim12i 568	. . . . . 6 |- ( ( F : ( 0..^ M ) --> X /\ P : ( 0..^ N ) --> Y ) -> ( F Fn ( 0..^ M ) /\ P Fn ( 0..^ N ) ) )
72	71	adantl 467	. . . . 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 467	. . . 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 5988	. . . 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 17	. . 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 2620	. . . . . 6 |- ( M =/= 0 <-> -. M = 0 )
77		elnnne0 10883	. . . . . . . 8 |- ( M e. NN <-> ( M e. NN0 /\ M =/= 0 ) )
78		0zd 10949	. . . . . . . . . . . . . . 15 |- ( M e. NN -> 0 e. ZZ )
79		nnz 10959	. . . . . . . . . . . . . . 15 |- ( M e. NN -> M e. ZZ )
80		nngt0 10638	. . . . . . . . . . . . . . 15 |- ( M e. NN -> 0 < M )
81	78, 79, 80	3jca 1185	. . . . . . . . . . . . . 14 |- ( M e. NN -> ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) )
82	81	adantr 466	. . . . . . . . . . . . 13 |- ( ( M e. NN /\ N e. NN0 ) -> ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) )
83		fzoopth 38751	. . . . . . . . . . . . 13 |- ( ( 0 e. ZZ /\ M e. ZZ /\ 0 < M ) -> ( ( 0..^ M ) = ( 0..^ N ) <-> ( 0 = 0 /\ M = N ) ) )
84	82, 83	syl 17	. . . . . . . . . . . 12 |- ( ( M e. NN /\ N e. NN0 ) -> ( ( 0..^ M ) = ( 0..^ N ) <-> ( 0 = 0 /\ M = N ) ) )
85		simpr 462	. . . . . . . . . . . 12 |- ( ( 0 = 0 /\ M = N ) -> M = N )
86	84, 85	syl6bi 231	. . . . . . . . . . 11 |- ( ( M e. NN /\ N e. NN0 ) -> ( ( 0..^ M ) = ( 0..^ N ) -> M = N ) )
87	86	anim1d 566	. . . . . . . . . 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 6309	. . . . . . . . . . 11 |- ( M = N -> ( 0..^ M ) = ( 0..^ N ) )
89	88	anim1i 570	. . . . . . . . . 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 206	. . . . . . . . 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 435	. . . . . . . 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 216	. . . . . . 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 441	. . . . . 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 221	. . . . 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 466	. . . 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 431	. . 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 256	. 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 797	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 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   =/= wne 2618   A.wral 2775   (/)c0 3761   class class class wbr 4420   Fn wfn 5592   -->wf 5593   ` cfv 5597  (class class class)co 6301   0cc0 9539   < clt 9675   <_ cle 9676   NNcn 10609   NN0cn0 10869   ZZcz 10937  ..^cfzo 11915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916

This theorem is referenced by: (None)