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Theorem pwfseqlem5 9062
 Description: Lemma for pwfseq 9063. Although in some ways pwfseqlem4 9061 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection from the set of finite sequences on an infinite set to . Now this alone would not be difficult to prove; this is mostly the claim of fseqen 8429. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas. If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 8413. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 8171), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 8416). (Contributed by Mario Carneiro, 31-May-2015.)
Hypotheses
Ref Expression
pwfseqlem5.g
pwfseqlem5.x
pwfseqlem5.h
pwfseqlem5.ps
pwfseqlem5.n
pwfseqlem5.o
pwfseqlem5.t
pwfseqlem5.p
pwfseqlem5.s
pwfseqlem5.q
pwfseqlem5.i
pwfseqlem5.k
Assertion
Ref Expression
pwfseqlem5
Distinct variable groups:   ,,   ,,,   ,,,P   ,,,,,,,,,   ,,,,,,,   ,,   N,   ,,,,   S,,   ,,,,   O,,,,,

Proof of Theorem pwfseqlem5
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwfseqlem5.g . 2
2 pwfseqlem5.x . 2
3 pwfseqlem5.h . 2
4 pwfseqlem5.ps . 2
5 vex 3112 . . . . . . . . . . 11
6 simprl3 1043 . . . . . . . . . . . 12
74, 6sylan2b 475 . . . . . . . . . . 11
8 pwfseqlem5.o . . . . . . . . . . . 12
98oiiso 7983 . . . . . . . . . . 11
105, 7, 9sylancr 663 . . . . . . . . . 10
11 isof1o 6221 . . . . . . . . . 10
1210, 11syl 16 . . . . . . . . 9
138oion 7982 . . . . . . . . . . . . 13
145, 13ax-mp 5 . . . . . . . . . . . 12
1514a1i 11 . . . . . . . . . . 11
168oien 7984 . . . . . . . . . . . . 13
175, 7, 16sylancr 663 . . . . . . . . . . . 12
181adantr 465 . . . . . . . . . . . . . . . 16
19 omex 8081 . . . . . . . . . . . . . . . . 17
20 ovex 6324 . . . . . . . . . . . . . . . . 17
2119, 20iunex 6780 . . . . . . . . . . . . . . . 16
22 f1dmex 6770 . . . . . . . . . . . . . . . 16
2318, 21, 22sylancl 662 . . . . . . . . . . . . . . 15
24 pwexb 6611 . . . . . . . . . . . . . . 15
2523, 24sylibr 212 . . . . . . . . . . . . . 14
26 simprl1 1041 . . . . . . . . . . . . . . 15
274, 26sylan2b 475 . . . . . . . . . . . . . 14
28 ssdomg 7581 . . . . . . . . . . . . . 14
2925, 27, 28sylc 60 . . . . . . . . . . . . 13
30 canth2g 7691 . . . . . . . . . . . . . 14
31 sdomdom 7563 . . . . . . . . . . . . . 14
3225, 30, 313syl 20 . . . . . . . . . . . . 13
33 domtr 7588 . . . . . . . . . . . . 13
3429, 32, 33syl2anc 661 . . . . . . . . . . . 12
35 endomtr 7593 . . . . . . . . . . . 12
3617, 34, 35syl2anc 661 . . . . . . . . . . 11
37 elharval 8010 . . . . . . . . . . 11
3815, 36, 37sylanbrc 664 . . . . . . . . . 10
39 pwfseqlem5.n . . . . . . . . . . 11
4039adantr 465 . . . . . . . . . 10
41 cardom 8388 . . . . . . . . . . . 12
42 simprr 757 . . . . . . . . . . . . . . 15
434, 42sylan2b 475 . . . . . . . . . . . . . 14
4417ensymd 7586 . . . . . . . . . . . . . 14
45 domentr 7594 . . . . . . . . . . . . . 14
4643, 44, 45syl2anc 661 . . . . . . . . . . . . 13
47 omelon 8084 . . . . . . . . . . . . . . 15
48 onenon 8351 . . . . . . . . . . . . . . 15
4947, 48ax-mp 5 . . . . . . . . . . . . . 14
50 onenon 8351 . . . . . . . . . . . . . . 15
5114, 50mp1i 12 . . . . . . . . . . . . . 14
52 carddom2 8379 . . . . . . . . . . . . . 14
5349, 51, 52sylancr 663 . . . . . . . . . . . . 13
5446, 53mpbird 232 . . . . . . . . . . . 12
5541, 54syl5eqssr 3548 . . . . . . . . . . 11
56 cardonle 8359 . . . . . . . . . . . 12
5715, 56syl 16 . . . . . . . . . . 11
5855, 57sstrd 3513 . . . . . . . . . 10
59 sseq2 3525 . . . . . . . . . . . 12
60 fveq2 5871 . . . . . . . . . . . . . 14
61 f1oeq1 5812 . . . . . . . . . . . . . 14
6260, 61syl 16 . . . . . . . . . . . . 13
63 xpeq12 5023 . . . . . . . . . . . . . . 15
6463anidms 645 . . . . . . . . . . . . . 14
65 f1oeq2 5813 . . . . . . . . . . . . . 14
6664, 65syl 16 . . . . . . . . . . . . 13
67 f1oeq3 5814 . . . . . . . . . . . . 13
6862, 66, 673bitrd 279 . . . . . . . . . . . 12
6959, 68imbi12d 320 . . . . . . . . . . 11
7069rspcv 3206 . . . . . . . . . 10
7138, 40, 58, 70syl3c 61 . . . . . . . . 9
72 f1oco 5843 . . . . . . . . 9
7312, 71, 72syl2anc 661 . . . . . . . 8
74 f1of 5821 . . . . . . . . . . . . . . 15
7512, 74syl 16 . . . . . . . . . . . . . 14
7675feqmptd 5926 . . . . . . . . . . . . 13
77 f1oeq1 5812 . . . . . . . . . . . . 13
7876, 77syl 16 . . . . . . . . . . . 12
7912, 78mpbid 210 . . . . . . . . . . 11
8075feqmptd 5926 . . . . . . . . . . . . 13
81 f1oeq1 5812 . . . . . . . . . . . . 13
8280, 81syl 16 . . . . . . . . . . . 12
8312, 82mpbid 210 . . . . . . . . . . 11
8479, 83xpf1o 7699 . . . . . . . . . 10
85 pwfseqlem5.t . . . . . . . . . . 11
86 f1oeq1 5812 . . . . . . . . . . 11
8785, 86ax-mp 5 . . . . . . . . . 10
8884, 87sylibr 212 . . . . . . . . 9
89 f1ocnv 5833 . . . . . . . . 9
9088, 89syl 16 . . . . . . . 8
91 f1oco 5843 . . . . . . . 8
9273, 90, 91syl2anc 661 . . . . . . 7
93 pwfseqlem5.p . . . . . . . 8
94 f1oeq1 5812 . . . . . . . 8
9593, 94ax-mp 5 . . . . . . 7
9692, 95sylibr 212 . . . . . 6
97 f1of1 5820 . . . . . 6
9896, 97syl 16 . . . . 5
99 f1of1 5820 . . . . . . . . . . . . 13
10012, 99syl 16 . . . . . . . . . . . 12
101 f1ssres 5793 . . . . . . . . . . . 12
102100, 58, 101syl2anc 661 . . . . . . . . . . 11
103 f1f1orn 5832 . . . . . . . . . . 11
104102, 103syl 16 . . . . . . . . . 10
10575, 58feqresmpt 5927 . . . . . . . . . . 11
106 f1oeq1 5812 . . . . . . . . . . 11
107105, 106syl 16 . . . . . . . . . 10
108104, 107mpbid 210 . . . . . . . . 9
109 mptresid 5333 . . . . . . . . . 10
110 f1oi 5856 . . . . . . . . . . 11
111 f1oeq1 5812 . . . . . . . . . . 11
112110, 111mpbiri 233 . . . . . . . . . 10
113109, 112mp1i 12 . . . . . . . . 9
114108, 113xpf1o 7699 . . . . . . . 8
115 pwfseqlem5.i . . . . . . . . 9
116 f1oeq1 5812 . . . . . . . . 9
117115, 116ax-mp 5 . . . . . . . 8
118114, 117sylibr 212 . . . . . . 7
119 f1of1 5820 . . . . . . 7
120118, 119syl 16 . . . . . 6
121 f1f 5786 . . . . . . 7
122 frn 5742 . . . . . . 7
123 xpss1 5116 . . . . . . 7
124102, 121, 122, 1234syl 21 . . . . . 6
125 f1ss 5791 . . . . . 6
126120, 124, 125syl2anc 661 . . . . 5
127 f1co 5795 . . . . 5
12898, 126, 127syl2anc 661 . . . 4
1295a1i 11 . . . . 5
130 peano1 6719 . . . . . . . 8
131130a1i 11 . . . . . . 7
13258, 131sseldd 3504 . . . . . 6
13375, 132ffvelrnd 6032 . . . . 5
134 pwfseqlem5.s . . . . 5
135 pwfseqlem5.q . . . . 5
136129, 133, 96, 134, 135fseqenlem2 8427 . . . 4
137 f1co 5795 . . . 4
138128, 136, 137syl2anc 661 . . 3
139 pwfseqlem5.k . . . 4
140 f1eq1 5781 . . . 4
141139, 140ax-mp 5 . . 3
142138, 141sylibr 212 . 2
143 eqid 2457 . 2
144 eqid 2457 . 2
145 eqid 2457 . . 3
146145fpwwe2cbv 9029 . 2
147 eqid 2457 . 2
1481, 2, 3, 4, 142, 143, 144, 146, 147pwfseqlem4 9061 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  {crab 2811   cvv 3109  [.wsbc 3327  i^icin 3474  C_wss 3475   c0 3784  ifcif 3941  ~Pcpw 4012  {csn 4029  <.cop 4035  U.cuni 4249  |^|cint 4286  U_ciun 4330   class class class wbr 4452  {copab 4509  e.cmpt 4510   cep 4794   cid 4795  Wewwe 4842   con0 4883  succsuc 4885  X.cxp 5002  'ccnv 5003  domcdm 5004  rancrn 5005  |cres 5006  "cima 5007  o.ccom 5008  -->wf 5589  -1-1->wf1 5590  -1-1-onto->wf1o 5592  `cfv 5593  Isomwiso 5594  (class class class)co 6296  e.cmpt2 6298   com 6700  seqomcseqom 7131   cmap 7439   cen 7533   cdom 7534   csdm 7535   cfn 7536  OrdIsocoi 7955   char 8003   ccrd 8337 This theorem is referenced by:  pwfseq  9063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-inf2 8079 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 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-int 4287  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6257  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6701  df-1st 6800  df-2nd 6801  df-recs 7061  df-rdg 7095  df-seqom 7132  df-1o 7149  df-er 7330  df-map 7441  df-en 7537  df-dom 7538  df-sdom 7539  df-fin 7540  df-oi 7956  df-har 8005  df-card 8341
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