Theorem rspceov 7466

Description: A frequently used special case of rspc2ev 3592 for operation values. (Contributed by NM, 21-Mar-2007.)

Assertion

Ref	Expression
rspceov	⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 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

Step	Hyp	Ref	Expression
1		oveq1 7424	. . 3 ⊢ (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦))
2	1	eqeq2d 2773	. 2 ⊢ (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦)))
3		oveq2 7425	. . 3 ⊢ (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷))
4	3	eqeq2d 2773	. 2 ⊢ (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷)))
5	2, 4	rspc2ev 3592	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  (class class class)co 7417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420

This theorem is used by:  iunfictbso  10121  genpprecl  11014  elz2  12637  zaddcl  12662  znq  13005  qaddcl  13019  qmulcl  13021  qreccl  13023  xpsff1o  17659  mndpfoOLD  18868  gafo  19429  lsmelvalix  19774  lsmelvalmi  19785  elmgplsmd  20292  evthicc2  25694  i1fadd  25929  i1fmul  25930  nnzsubs  28658  nnzs  28659  0zs  28661  zmulscld  28670  elzn0s  28671  2clwwlk2clwwlk  30838  isgrpoi  30987  shscli  31806  shsva  31809  shunssi  31857  pjpjhth  31914  spanunsni  32068  pjjsi  32189  ofrn2  33121  pstmfval  34414  ismblfin  38418  itg2addnc  38431  blbnd  38545  isgrpda  38713  sbgoldbalt  48705