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Theorem frsucmptn 7174
 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 7173 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 5880 . 2
3 frfnom 7170 . . . . . 6
4 fndm 5685 . . . . . 6
53, 4ax-mp 5 . . . . 5
65eleq2i 2541 . . . 4
7 peano2b 6727 . . . . 5
8 frsuc 7172 . . . . . . . 8
91fveq1i 5880 . . . . . . . . 9
109fveq2i 5882 . . . . . . . 8
118, 10syl6eqr 2523 . . . . . . 7
12 nfmpt1 4485 . . . . . . . . . . . 12
13 frsucmpt.1 . . . . . . . . . . . 12
1412, 13nfrdg 7150 . . . . . . . . . . 11
15 nfcv 2612 . . . . . . . . . . 11
1614, 15nfres 5113 . . . . . . . . . 10
171, 16nfcxfr 2610 . . . . . . . . 9
18 frsucmpt.2 . . . . . . . . 9
1917, 18nffv 5886 . . . . . . . 8
20 frsucmpt.3 . . . . . . . 8
21 frsucmpt.5 . . . . . . . 8
22 eqid 2471 . . . . . . . 8
2319, 20, 21, 22fvmptnf 5982 . . . . . . 7
2411, 23sylan9eqr 2527 . . . . . 6
2524ex 441 . . . . 5
267, 25syl5bir 226 . . . 4
276, 26syl5bi 225 . . 3
28 ndmfv 5903 . . 3
2927, 28pm2.61d1 164 . 2
302, 29syl5eq 2517 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wceq 1452   wcel 1904  wnfc 2599  cvv 3031  c0 3722   cmpt 4454   cdm 4839   cres 4841   csuc 5432   wfn 5584  cfv 5589  com 6711  crdg 7145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146 This theorem is referenced by:  trpredlem1  30539
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