Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqeq2 Structured version   Unicode version

Theorem seqeq2 11332
 Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq2

Proof of Theorem seqeq2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2439 . . . . 5
2 oveq 6090 . . . . . 6
32opeq2d 3993 . . . . 5
41, 1, 3mpt2eq123dv 6139 . . . 4
5 rdgeq1 6672 . . . 4
64, 5syl 16 . . 3
76imaeq1d 5205 . 2
8 df-seq 11329 . 2
9 df-seq 11329 . 2
107, 8, 93eqtr4g 2495 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653  cvv 2958  cop 3819  com 4848  cima 4884  cfv 5457  (class class class)co 6084   cmpt2 6086  crdg 6670  c1 8996   caddc 8998   cseq 11328 This theorem is referenced by:  seqeq2d  11335  sadcom  12980  gxfval  21850  ressmulgnn  24210  cvmliftlem15  24990 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-recs 6636  df-rdg 6671  df-seq 11329
 Copyright terms: Public domain W3C validator