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Theorem ovrspc2v 7156
Description: If an operation value is element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.)
Assertion
Ref Expression
ovrspc2v (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑦,𝑌   𝑥,𝑋,𝑦
Allowed substitution hint:   𝑌(𝑥)

Proof of Theorem ovrspc2v
StepHypRef Expression
1 oveq1 7137 . . 3 (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
21eleq1d 2896 . 2 (𝑥 = 𝑋 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑦) ∈ 𝐶))
3 oveq2 7138 . . 3 (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
43eleq1d 2896 . 2 (𝑦 = 𝑌 → ((𝑋𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑌) ∈ 𝐶))
52, 4rspc2va 3611 1 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3126  (class class class)co 7130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-ov 7133
This theorem is referenced by:  off  7399  mgmcl  17834  mndpropd  17915  issubmnd  17917  submcl  17956  issubg2  18273  gass  18410  lmodprop2d  19672  lsspropd  19765  gsummatr01lem2  21241  off2  30375  ofcf  31370  submgmcl  44233  clcllaw  44270
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