MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rspceov Structured version   Visualization version   GIF version

Theorem rspceov 6677
Description: A frequently used special case of rspc2ev 3319 for operation values. (Contributed by NM, 21-Mar-2007.)
Assertion
Ref Expression
rspceov ((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑦,𝐷   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐷(𝑥)

Proof of Theorem rspceov
StepHypRef Expression
1 oveq1 6642 . . 3 (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦))
21eqeq2d 2630 . 2 (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦)))
3 oveq2 6643 . . 3 (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷))
43eqeq2d 2630 . 2 (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷)))
52, 4rspc2ev 3319 1 ((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  wrex 2910  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638
This theorem is referenced by:  iunfictbso  8922  genpprecl  9808  elz2  11379  zaddcl  11402  znq  11777  qaddcl  11789  qmulcl  11791  qreccl  11793  xpsff1o  16209  mndpfo  17295  gafo  17710  lsmelvalix  18037  lsmelvalmi  18048  evthicc2  23210  i1fadd  23443  i1fmul  23444  isgrpoi  27322  shscli  28146  shsva  28149  shunssi  28197  pjpjhth  28254  spanunsni  28408  pjjsi  28529  ofrn2  29415  pstmfval  29913  ismblfin  33421  itg2addnc  33435  blbnd  33557  isgrpda  33725  sbgoldbalt  41434
  Copyright terms: Public domain W3C validator