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Theorem frsucmptn 6688
 Description: The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class is a proper class). This is a technical lemma that can be used together with frsucmpt 6687 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
frsucmpt.1
frsucmpt.2
frsucmpt.3
frsucmpt.4
frsucmpt.5
Assertion
Ref Expression
frsucmptn

Proof of Theorem frsucmptn
StepHypRef Expression
1 frsucmpt.4 . . 3
21fveq1i 5721 . 2
3 frfnom 6684 . . . . . 6
4 fndm 5536 . . . . . 6
53, 4ax-mp 8 . . . . 5
65eleq2i 2499 . . . 4
7 peano2b 4853 . . . . 5
8 frsuc 6686 . . . . . . . 8
91fveq1i 5721 . . . . . . . . 9
109fveq2i 5723 . . . . . . . 8
118, 10syl6eqr 2485 . . . . . . 7
12 nfmpt1 4290 . . . . . . . . . . . 12
13 frsucmpt.1 . . . . . . . . . . . 12
1412, 13nfrdg 6664 . . . . . . . . . . 11
15 nfcv 2571 . . . . . . . . . . 11
1614, 15nfres 5140 . . . . . . . . . 10
171, 16nfcxfr 2568 . . . . . . . . 9
18 frsucmpt.2 . . . . . . . . 9
1917, 18nffv 5727 . . . . . . . 8
20 frsucmpt.3 . . . . . . . 8
21 frsucmpt.5 . . . . . . . 8
22 eqid 2435 . . . . . . . 8
2319, 20, 21, 22fvmptnf 5814 . . . . . . 7
2411, 23sylan9eqr 2489 . . . . . 6
2524ex 424 . . . . 5
267, 25syl5bir 210 . . . 4
276, 26syl5bi 209 . . 3
28 ndmfv 5747 . . 3
2927, 28pm2.61d1 153 . 2
302, 29syl5eq 2479 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wceq 1652   wcel 1725  wnfc 2558  cvv 2948  c0 3620   cmpt 4258   csuc 4575  com 4837   cdm 4870   cres 4872   wfn 5441  cfv 5446  crdg 6659 This theorem is referenced by:  trpredlem1  25497 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660
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