Theorem rspceov 6590

Description: A frequently used special case of rspc2ev 3295 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 6556	. . 3 ⊢ (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦))
2	1	eqeq2d 2620	. 2 ⊢ (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦)))
3		oveq2 6557	. . 3 ⊢ (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷))
4	3	eqeq2d 2620	. 2 ⊢ (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷)))
5	2, 4	rspc2ev 3295	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦))

Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by:  iunfictbso  8820  genpprecl  9702  elz2  11271  zaddcl  11294  znq  11668  qaddcl  11680  qmulcl  11682  qreccl  11684  xpsff1o  16051  mndpfo  17137  gafo  17552  lsmelvalix  17879  lsmelvalmi  17890  evthicc2  23036  i1fadd  23268  i1fmul  23269  isgrpoi  26736  shscli  27560  shsva  27563  shunssi  27611  pjpjhth  27668  spanunsni  27822  pjjsi  27943  ofrn2  28822  pstmfval  29267  ismblfin  32620  itg2addnc  32634  blbnd  32756  isgrpda  32924  sgoldbalt  40203