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Theorem cantnfp1OLD 8160
 Description: If is created by adding a single term to , where is larger than any element of the support of , then is also a finitely supported function and it is assigned the value where is the value of . (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of cantnfp1 8134 as of 1-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1 CNF
cantnfsOLD.2
cantnfsOLD.3
cantnfp1OLD.4
cantnfp1OLD.5
cantnfp1OLD.6
cantnfp1OLD.7
cantnfp1OLD.f
Assertion
Ref Expression
cantnfp1OLD CNF CNF
Distinct variable groups:   ,   ,   ,   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem cantnfp1OLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfp1OLD.f . . . . . 6
2 eqeq1 2408 . . . . . . . 8
3 eqeq1 2408 . . . . . . . 8
4 cantnfsOLD.3 . . . . . . . . . . . . 13
5 cantnfp1OLD.5 . . . . . . . . . . . . 13
6 onelon 5437 . . . . . . . . . . . . 13
74, 5, 6syl2anc 661 . . . . . . . . . . . 12
8 eloni 5422 . . . . . . . . . . . 12
9 ordirr 5430 . . . . . . . . . . . 12
107, 8, 93syl 18 . . . . . . . . . . 11
11 fvex 5861 . . . . . . . . . . . . . 14
12 dif1o 7189 . . . . . . . . . . . . . 14
1311, 12mpbiran 921 . . . . . . . . . . . . 13
14 cantnfp1OLD.4 . . . . . . . . . . . . . . . . . 18
15 cantnfsOLD.1 . . . . . . . . . . . . . . . . . . 19 CNF
16 cantnfsOLD.2 . . . . . . . . . . . . . . . . . . 19
1715, 16, 4cantnfsOLD 8149 . . . . . . . . . . . . . . . . . 18
1814, 17mpbid 212 . . . . . . . . . . . . . . . . 17
1918simpld 459 . . . . . . . . . . . . . . . 16
20 ffn 5716 . . . . . . . . . . . . . . . 16
21 elpreima 5987 . . . . . . . . . . . . . . . 16
2219, 20, 213syl 18 . . . . . . . . . . . . . . 15
23 cantnfp1OLD.7 . . . . . . . . . . . . . . . 16
2423sseld 3443 . . . . . . . . . . . . . . 15
2522, 24sylbird 237 . . . . . . . . . . . . . 14
265, 25mpand 675 . . . . . . . . . . . . 13
2713, 26syl5bir 220 . . . . . . . . . . . 12
2827necon1bd 2623 . . . . . . . . . . 11
2910, 28mpd 15 . . . . . . . . . 10
3029ad3antrrr 730 . . . . . . . . 9
31 simpr 461 . . . . . . . . . 10
3231fveq2d 5855 . . . . . . . . 9
33 simpllr 763 . . . . . . . . 9
3430, 32, 333eqtr4rd 2456 . . . . . . . 8
35 eqidd 2405 . . . . . . . 8
362, 3, 34, 35ifbothda 3922 . . . . . . 7
3736mpteq2dva 4483 . . . . . 6
381, 37syl5eq 2457 . . . . 5
3919feqmptd 5904 . . . . . 6
4039adantr 465 . . . . 5
4138, 40eqtr4d 2448 . . . 4
4214adantr 465 . . . 4
4341, 42eqeltrd 2492 . . 3
44 oecl 7226 . . . . . . . 8
4516, 4, 44syl2anc 661 . . . . . . 7
4615, 16, 4cantnff 8127 . . . . . . . 8 CNF
4746, 14ffvelrnd 6012 . . . . . . 7 CNF
48 onelon 5437 . . . . . . 7 CNF CNF
4945, 47, 48syl2anc 661 . . . . . 6 CNF
5049adantr 465 . . . . 5 CNF
51 oa0r 7227 . . . . 5 CNF CNF CNF
5250, 51syl 17 . . . 4 CNF CNF
53 oveq2 6288 . . . . . 6
54 oecl 7226 . . . . . . . 8
5516, 7, 54syl2anc 661 . . . . . . 7
56 om0 7206 . . . . . . 7
5755, 56syl 17 . . . . . 6
5853, 57sylan9eqr 2467 . . . . 5
5958oveq1d 6295 . . . 4 CNF CNF
6041fveq2d 5855 . . . 4 CNF CNF
6152, 59, 603eqtr4rd 2456 . . 3 CNF CNF
6243, 61jca 532 . 2 CNF CNF
6316adantr 465 . . . 4
644adantr 465 . . . 4
6514adantr 465 . . . 4
665adantr 465 . . . 4
67 cantnfp1OLD.6 . . . . 5
6867adantr 465 . . . 4
6923adantr 465 . . . 4
7015, 63, 64, 65, 66, 68, 69, 1cantnfp1lem1OLD 8157 . . 3
71 onelon 5437 . . . . . . 7
7216, 67, 71syl2anc 661 . . . . . 6
73 on0eln0 5467 . . . . . 6
7472, 73syl 17 . . . . 5
7574biimpar 485 . . . 4
76 eqid 2404 . . . 4 OrdIso OrdIso
77 eqid 2404 . . . 4 seq𝜔 OrdIso OrdIso seq𝜔 OrdIso OrdIso
78 eqid 2404 . . . 4 OrdIso OrdIso
79 eqid 2404 . . . 4 seq𝜔 OrdIso OrdIso seq𝜔 OrdIso OrdIso
8015, 63, 64, 65, 66, 68, 69, 1, 75, 76, 77, 78, 79cantnfp1lem3OLD 8159 . . 3 CNF CNF
8170, 80jca 532 . 2 CNF CNF
8262, 81pm2.61dane 2723 1 CNF CNF
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 186   wa 369   wceq 1407   wcel 1844   wne 2600  cvv 3061   cdif 3413   wss 3416  c0 3740  cif 3887   cmpt 4455   cep 4734  ccnv 4824   cdm 4825  cima 4828   word 5411  con0 5412   wfn 5566  wf 5567  cfv 5571  (class class class)co 6280   cmpt2 6282  seq𝜔cseqom 7151  c1o 7162   coa 7166   comu 7167   coe 7168  cfn 7556  OrdIsocoi 7970   CNF ccnf 8112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576 This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-seqom 7152  df-1o 7169  df-2o 7170  df-oadd 7173  df-omul 7174  df-oexp 7175  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-oi 7971  df-cnf 8113 This theorem is referenced by:  cantnflem1dOLD  8164  cantnflem1OLD  8165  cantnflem3OLD  8166
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