Theorem rspceov 6836

Description: A frequently used special case of rspc2ev 3472 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 6799	. . 3 ⊢ (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦))
2	1	eqeq2d 2780	. 2 ⊢ (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦)))
3		oveq2 6800	. . 3 ⊢ (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷))
4	3	eqeq2d 2780	. 2 ⊢ (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷)))
5	2, 4	rspc2ev 3472	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦))

Syntax hints:   → wi 4   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  ∃wrex 3061  (class class class)co 6792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795

This theorem is referenced by:  iunfictbso  9136  genpprecl  10024  elz2  11595  zaddcl  11618  znq  11994  qaddcl  12006  qmulcl  12008  qreccl  12010  xpsff1o  16435  mndpfo  17521  gafo  17935  lsmelvalix  18262  lsmelvalmi  18273  evthicc2  23447  i1fadd  23681  i1fmul  23682  2clwwlk2clwwlk  27532  isgrpoi  27686  shscli  28510  shsva  28513  shunssi  28561  pjpjhth  28618  spanunsni  28772  pjjsi  28893  ofrn2  29776  pstmfval  30273  ismblfin  33776  itg2addnc  33789  blbnd  33911  isgrpda  34079  sbgoldbalt  42187