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Theorem seqeq3 11320
 Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq3

Proof of Theorem seqeq3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2436 . . . . 5
2 fveq1 5719 . . . . . . 7
32oveq2d 6089 . . . . . 6
43opeq2d 3983 . . . . 5
51, 1, 4mpt2eq123dv 6128 . . . 4
6 fveq1 5719 . . . . 5
76opeq2d 3983 . . . 4
8 rdgeq12 6663 . . . 4
95, 7, 8syl2anc 643 . . 3
109imaeq1d 5194 . 2
11 df-seq 11316 . 2
12 df-seq 11316 . 2
1310, 11, 123eqtr4g 2492 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652  cvv 2948  cop 3809  com 4837  cima 4873  cfv 5446  (class class class)co 6073   cmpt2 6075  crdg 6659  c1 8983   caddc 8985   cseq 11315 This theorem is referenced by:  seqeq3d  11323  geolim3  20248  leibpilem2  20773  basel  20864  cbvprod  25233  iprodmul  25308  faclim  25357  ovoliunnfl  26238  voliunnfl  26240  heiborlem10  26520 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-seq 11316
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