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Theorem cantnfsOLD 8136
 Description: Elementhood in the set of finitely supported functions from to . (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnfs 8106 as of 28-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1
cantnfsOLD.2
cantnfsOLD.3
Assertion
Ref Expression
cantnfsOLD

Proof of Theorem cantnfsOLD
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cantnfsOLD.1 . . . . 5
2 eqid 2457 . . . . . 6
3 cantnfsOLD.2 . . . . . 6
4 cantnfsOLD.3 . . . . . 6
52, 3, 4cantnfdmOLD 8104 . . . . 5
61, 5syl5eq 2510 . . . 4
76eleq2d 2527 . . 3
8 cnveq 5181 . . . . . 6
98imaeq1d 5341 . . . . 5
109eleq1d 2526 . . . 4
1110elrab 3257 . . 3
127, 11syl6bb 261 . 2
13 elmapg 7452 . . . 4
143, 4, 13syl2anc 661 . . 3
1514anbi1d 704 . 2
1612, 15bitrd 253 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  {crab 2811   cvv 3109  \cdif 3472   con0 4883  'ccnv 5003  domcdm 5004  "cima 5007  -->`wf 5589  (class class class)co 6296   c1o 7142   cmap 7439   cfn 7536   ccnf 8099 This theorem is referenced by:  cantnfclOLD  8137  cantnfleOLD  8141  cantnfltOLD  8142  cantnfp1lem1OLD  8144  cantnfp1lem2OLD  8145  cantnfp1lem3OLD  8146  cantnfp1OLD  8147  cantnflem1aOLD  8148  cantnflem1bOLD  8149  cantnflem1cOLD  8150  cantnflem1dOLD  8151  cantnflem1OLD  8152  cantnflem3OLD  8153  cantnfOLD  8155  cnfcomlemOLD  8172  cnfcomOLD  8173  cnfcom2lemOLD  8174  cnfcom3lemOLD  8176  cnfcom3OLD  8177 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 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  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-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  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-riota 6257  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801  df-supp 6919  df-recs 7061  df-rdg 7095  df-seqom 7132  df-1o 7149  df-map 7441  df-fsupp 7850  df-oi 7956  df-cnf 8100
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