Theorem erdszelem9 31181

Description: Lemma for erdsze 31184. (Contributed by Mario Carneiro, 22-Jan-2015.)

Hypotheses

Ref	Expression
erdsze.n	|- ( ph -> N e. NN )
erdsze.f	|- ( ph -> F : ( 1 ... N ) -1-1-> RR )
erdszelem.i	|- I = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , < ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
erdszelem.j	|- J = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , `' < ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
erdszelem.t	|- T = ( n e. ( 1 ... N ) |-> <. ( I ` n ) , ( J ` n ) >. )

Assertion

Ref	Expression
erdszelem9	|- ( ph -> T : ( 1 ... N ) -1-1-> ( NN X. NN ) )

Distinct variable groups:   x, y, n, F   n, I, x, y   n, J, x, y   n, N, x, y   ph, n, x, y

Allowed substitution hints:   T( x, y, n)

Proof of Theorem erdszelem9

Dummy variables w z a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		erdsze.n	. . . . . 6 |- ( ph -> N e. NN )
2		erdsze.f	. . . . . 6 |- ( ph -> F : ( 1 ... N ) -1-1-> RR )
3		erdszelem.i	. . . . . 6 |- I = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , < ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
4		ltso 10118	. . . . . 6 |- < Or RR
5	1, 2, 3, 4	erdszelem6 31178	. . . . 5 |- ( ph -> I : ( 1 ... N ) --> NN )
6	5	ffvelrnda 6359	. . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( I ` n ) e. NN )
7		erdszelem.j	. . . . . 6 |- J = ( x e. ( 1 ... N ) |-> sup ( ( # " { y e. ~P ( 1 ... x ) | ( ( F |` y ) Isom < , `' < ( y , ( F " y ) ) /\ x e. y ) } ) , RR , < ) )
8		gtso 10119	. . . . . 6 |- `' < Or RR
9	1, 2, 7, 8	erdszelem6 31178	. . . . 5 |- ( ph -> J : ( 1 ... N ) --> NN )
10	9	ffvelrnda 6359	. . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( J ` n ) e. NN )
11		opelxpi 5148	. . . 4 |- ( ( ( I ` n ) e. NN /\ ( J ` n ) e. NN ) -> <. ( I ` n ) , ( J ` n ) >. e. ( NN X. NN ) )
12	6, 10, 11	syl2anc 693	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> <. ( I ` n ) , ( J ` n ) >. e. ( NN X. NN ) )
13		erdszelem.t	. . 3 |- T = ( n e. ( 1 ... N ) |-> <. ( I ` n ) , ( J ` n ) >. )
14	12, 13	fmptd 6385	. 2 |- ( ph -> T : ( 1 ... N ) --> ( NN X. NN ) )
15		fveq2 6191	. . . . . 6 |- ( a = z -> ( T ` a ) = ( T ` z ) )
16		fveq2 6191	. . . . . 6 |- ( b = w -> ( T ` b ) = ( T ` w ) )
17	15, 16	eqeqan12d 2638	. . . . 5 |- ( ( a = z /\ b = w ) -> ( ( T ` a ) = ( T ` b ) <-> ( T ` z ) = ( T ` w ) ) )
18		eqeq12 2635	. . . . 5 |- ( ( a = z /\ b = w ) -> ( a = b <-> z = w ) )
19	17, 18	imbi12d 334	. . . 4 |- ( ( a = z /\ b = w ) -> ( ( ( T ` a ) = ( T ` b ) -> a = b ) <-> ( ( T ` z ) = ( T ` w ) -> z = w ) ) )
20		fveq2 6191	. . . . . . 7 |- ( a = w -> ( T ` a ) = ( T ` w ) )
21		fveq2 6191	. . . . . . 7 |- ( b = z -> ( T ` b ) = ( T ` z ) )
22	20, 21	eqeqan12d 2638	. . . . . 6 |- ( ( a = w /\ b = z ) -> ( ( T ` a ) = ( T ` b ) <-> ( T ` w ) = ( T ` z ) ) )
23		eqcom 2629	. . . . . 6 |- ( ( T ` w ) = ( T ` z ) <-> ( T ` z ) = ( T ` w ) )
24	22, 23	syl6bb 276	. . . . 5 |- ( ( a = w /\ b = z ) -> ( ( T ` a ) = ( T ` b ) <-> ( T ` z ) = ( T ` w ) ) )
25		eqeq12 2635	. . . . . 6 |- ( ( a = w /\ b = z ) -> ( a = b <-> w = z ) )
26		eqcom 2629	. . . . . 6 |- ( w = z <-> z = w )
27	25, 26	syl6bb 276	. . . . 5 |- ( ( a = w /\ b = z ) -> ( a = b <-> z = w ) )
28	24, 27	imbi12d 334	. . . 4 |- ( ( a = w /\ b = z ) -> ( ( ( T ` a ) = ( T ` b ) -> a = b ) <-> ( ( T ` z ) = ( T ` w ) -> z = w ) ) )
29		elfzelz 12342	. . . . . . 7 |- ( z e. ( 1 ... N ) -> z e. ZZ )
30	29	zred 11482	. . . . . 6 |- ( z e. ( 1 ... N ) -> z e. RR )
31	30	ssriv 3607	. . . . 5 |- ( 1 ... N ) C_ RR
32	31	a1i 11	. . . 4 |- ( ph -> ( 1 ... N ) C_ RR )
33		biidd 252	. . . 4 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) ) ) -> ( ( ( T ` z ) = ( T ` w ) -> z = w ) <-> ( ( T ` z ) = ( T ` w ) -> z = w ) ) )
34		simpr1 1067	. . . . . . . 8 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> z e. ( 1 ... N ) )
35		fveq2 6191	. . . . . . . . . 10 |- ( n = z -> ( I ` n ) = ( I ` z ) )
36		fveq2 6191	. . . . . . . . . 10 |- ( n = z -> ( J ` n ) = ( J ` z ) )
37	35, 36	opeq12d 4410	. . . . . . . . 9 |- ( n = z -> <. ( I ` n ) , ( J ` n ) >. = <. ( I ` z ) , ( J ` z ) >. )
38		opex 4932	. . . . . . . . 9 |- <. ( I ` z ) , ( J ` z ) >. e. _V
39	37, 13, 38	fvmpt 6282	. . . . . . . 8 |- ( z e. ( 1 ... N ) -> ( T ` z ) = <. ( I ` z ) , ( J ` z ) >. )
40	34, 39	syl 17	. . . . . . 7 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( T ` z ) = <. ( I ` z ) , ( J ` z ) >. )
41		simpr2 1068	. . . . . . . 8 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> w e. ( 1 ... N ) )
42		fveq2 6191	. . . . . . . . . 10 |- ( n = w -> ( I ` n ) = ( I ` w ) )
43		fveq2 6191	. . . . . . . . . 10 |- ( n = w -> ( J ` n ) = ( J ` w ) )
44	42, 43	opeq12d 4410	. . . . . . . . 9 |- ( n = w -> <. ( I ` n ) , ( J ` n ) >. = <. ( I ` w ) , ( J ` w ) >. )
45		opex 4932	. . . . . . . . 9 |- <. ( I ` w ) , ( J ` w ) >. e. _V
46	44, 13, 45	fvmpt 6282	. . . . . . . 8 |- ( w e. ( 1 ... N ) -> ( T ` w ) = <. ( I ` w ) , ( J ` w ) >. )
47	41, 46	syl 17	. . . . . . 7 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( T ` w ) = <. ( I ` w ) , ( J ` w ) >. )
48	40, 47	eqeq12d 2637	. . . . . 6 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( ( T ` z ) = ( T ` w ) <-> <. ( I ` z ) , ( J ` z ) >. = <. ( I ` w ) , ( J ` w ) >. ) )
49		fvex 6201	. . . . . . . 8 |- ( I ` z ) e. _V
50		fvex 6201	. . . . . . . 8 |- ( J ` z ) e. _V
51	49, 50	opth 4945	. . . . . . 7 |- ( <. ( I ` z ) , ( J ` z ) >. = <. ( I ` w ) , ( J ` w ) >. <-> ( ( I ` z ) = ( I ` w ) /\ ( J ` z ) = ( J ` w ) ) )
52	34, 30	syl 17	. . . . . . . . . 10 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> z e. RR )
53	31, 41	sseldi 3601	. . . . . . . . . 10 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> w e. RR )
54		simpr3 1069	. . . . . . . . . 10 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> z <_ w )
55	52, 53, 54	leltned 10190	. . . . . . . . 9 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( z < w <-> w =/= z ) )
56	2	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> F : ( 1 ... N ) -1-1-> RR )
57		f1fveq 6519	. . . . . . . . . . . . . . . . 17 |- ( ( F : ( 1 ... N ) -1-1-> RR /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) ) ) -> ( ( F ` z ) = ( F ` w ) <-> z = w ) )
58	56, 34, 41, 57	syl12anc 1324	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( ( F ` z ) = ( F ` w ) <-> z = w ) )
59	58, 26	syl6bbr 278	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( ( F ` z ) = ( F ` w ) <-> w = z ) )
60	59	necon3bid 2838	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( ( F ` z ) =/= ( F ` w ) <-> w =/= z ) )
61	55, 60	bitr4d 271	. . . . . . . . . . . . 13 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( z < w <-> ( F ` z ) =/= ( F ` w ) ) )
62	61	biimpa 501	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> ( F ` z ) =/= ( F ` w ) )
63		f1f 6101	. . . . . . . . . . . . . . . 16 |- ( F : ( 1 ... N ) -1-1-> RR -> F : ( 1 ... N ) --> RR )
64	2, 63	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> F : ( 1 ... N ) --> RR )
65	64	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> F : ( 1 ... N ) --> RR )
66	34	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> z e. ( 1 ... N ) )
67	65, 66	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> ( F ` z ) e. RR )
68	41	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> w e. ( 1 ... N ) )
69	65, 68	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> ( F ` w ) e. RR )
70	67, 69	lttri2d 10176	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> ( ( F ` z ) =/= ( F ` w ) <-> ( ( F ` z ) < ( F ` w ) \/ ( F ` w ) < ( F ` z ) ) ) )
71	62, 70	mpbid 222	. . . . . . . . . . 11 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> ( ( F ` z ) < ( F ` w ) \/ ( F ` w ) < ( F ` z ) ) )
72	1	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> N e. NN )
73	2	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> F : ( 1 ... N ) -1-1-> RR )
74		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> z < w )
75	72, 73, 3, 4, 66, 68, 74	erdszelem8 31180	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> ( ( I ` z ) = ( I ` w ) -> -. ( F ` z ) < ( F ` w ) ) )
76	72, 73, 7, 8, 66, 68, 74	erdszelem8 31180	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> ( ( J ` z ) = ( J ` w ) -> -. ( F ` z ) `' < ( F ` w ) ) )
77	75, 76	anim12d 586	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> ( ( ( I ` z ) = ( I ` w ) /\ ( J ` z ) = ( J ` w ) ) -> ( -. ( F ` z ) < ( F ` w ) /\ -. ( F ` z ) `' < ( F ` w ) ) ) )
78		ioran 511	. . . . . . . . . . . . 13 |- ( -. ( ( F ` z ) < ( F ` w ) \/ ( F ` w ) < ( F ` z ) ) <-> ( -. ( F ` z ) < ( F ` w ) /\ -. ( F ` w ) < ( F ` z ) ) )
79		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( F ` z ) e. _V
80		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( F ` w ) e. _V
81	79, 80	brcnv 5305	. . . . . . . . . . . . . . 15 |- ( ( F ` z ) `' < ( F ` w ) <-> ( F ` w ) < ( F ` z ) )
82	81	notbii 310	. . . . . . . . . . . . . 14 |- ( -. ( F ` z ) `' < ( F ` w ) <-> -. ( F ` w ) < ( F ` z ) )
83	82	anbi2i 730	. . . . . . . . . . . . 13 |- ( ( -. ( F ` z ) < ( F ` w ) /\ -. ( F ` z ) `' < ( F ` w ) ) <-> ( -. ( F ` z ) < ( F ` w ) /\ -. ( F ` w ) < ( F ` z ) ) )
84	78, 83	bitr4i 267	. . . . . . . . . . . 12 |- ( -. ( ( F ` z ) < ( F ` w ) \/ ( F ` w ) < ( F ` z ) ) <-> ( -. ( F ` z ) < ( F ` w ) /\ -. ( F ` z ) `' < ( F ` w ) ) )
85	77, 84	syl6ibr 242	. . . . . . . . . . 11 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> ( ( ( I ` z ) = ( I ` w ) /\ ( J ` z ) = ( J ` w ) ) -> -. ( ( F ` z ) < ( F ` w ) \/ ( F ` w ) < ( F ` z ) ) ) )
86	71, 85	mt2d 131	. . . . . . . . . 10 |- ( ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) /\ z < w ) -> -. ( ( I ` z ) = ( I ` w ) /\ ( J ` z ) = ( J ` w ) ) )
87	86	ex 450	. . . . . . . . 9 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( z < w -> -. ( ( I ` z ) = ( I ` w ) /\ ( J ` z ) = ( J ` w ) ) ) )
88	55, 87	sylbird 250	. . . . . . . 8 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( w =/= z -> -. ( ( I ` z ) = ( I ` w ) /\ ( J ` z ) = ( J ` w ) ) ) )
89	88	necon4ad 2813	. . . . . . 7 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( ( ( I ` z ) = ( I ` w ) /\ ( J ` z ) = ( J ` w ) ) -> w = z ) )
90	51, 89	syl5bi 232	. . . . . 6 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( <. ( I ` z ) , ( J ` z ) >. = <. ( I ` w ) , ( J ` w ) >. -> w = z ) )
91	48, 90	sylbid 230	. . . . 5 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( ( T ` z ) = ( T ` w ) -> w = z ) )
92	91, 26	syl6ib 241	. . . 4 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) /\ z <_ w ) ) -> ( ( T ` z ) = ( T ` w ) -> z = w ) )
93	19, 28, 32, 33, 92	wlogle 10561	. . 3 |- ( ( ph /\ ( z e. ( 1 ... N ) /\ w e. ( 1 ... N ) ) ) -> ( ( T ` z ) = ( T ` w ) -> z = w ) )
94	93	ralrimivva 2971	. 2 |- ( ph -> A. z e. ( 1 ... N ) A. w e. ( 1 ... N ) ( ( T ` z ) = ( T ` w ) -> z = w ) )
95		dff13 6512	. 2 |- ( T : ( 1 ... N ) -1-1-> ( NN X. NN ) <-> ( T : ( 1 ... N ) --> ( NN X. NN ) /\ A. z e. ( 1 ... N ) A. w e. ( 1 ... N ) ( ( T ` z ) = ( T ` w ) -> z = w ) ) )
96	14, 94, 95	sylanbrc 698	1 |- ( ph -> T : ( 1 ... N ) -1-1-> ( NN X. NN ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   C_ wss 3574   ~Pcpw 4158   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   -->wf 5884   -1-1->wf1 5885   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   supcsup 8346   RRcr 9935   1c1 9937   < clt 10074   <_ cle 10075   NNcn 11020   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  erdszelem10  31182