Theorem rspceov 6037

Description: A frequently used special case of rspc2ev 2922 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 6001	. . 3 ⊢ (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦))
2	1	eqeq2d 2241	. 2 ⊢ (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦)))
3		oveq2 6002	. . 3 ⊢ (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷))
4	3	eqeq2d 2241	. 2 ⊢ (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷)))
5	2, 4	rspc2ev 2922	1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = (𝐶𝐹𝐷)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = (𝑥𝐹𝑦))

Syntax hints:   → wi 4   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  (class class class)co 5994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5274  df-fv 5322  df-ov 5997

This theorem is referenced by:  genpprecll  7689  genppreclu  7690  elz2  9506  znq  9807  qaddcl  9818  qmulcl  9820  qreccl  9825  xpsff1o  13368  mndpfo  13457