Theorem ovrspc2v 7424

Description: If an operation value is an 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	⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (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

Step	Hyp	Ref	Expression
1		oveq1 7405	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
2	1	eleq1d 2849	. 2 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑦) ∈ 𝐶))
3		oveq2 7406	. . 3 ⊢ (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
4	3	eleq1d 2849	. 2 ⊢ (𝑦 = 𝑌 → ((𝑋𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑌) ∈ 𝐶))
5	2, 4	rspc2va 3595	1 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  (class class class)co 7398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401

This theorem is referenced by:  off  7680  mgmcl  18679  submgmcl  18743  sgrppropd  18767  mndpropd  18795  issubmnd  18797  submcl  18848  issubg2  19185  gass  19343  lmodprop2d  20993  lsspropd  21086  gsummatr01lem2  22718  off2  32845  ofcf  34402  fsuppind  43177  clcllaw  48818