Theorem ovrspc2v 7457

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	⊢ (((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 7438	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
2	1	eleq1d 2826	. 2 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑦) ∈ 𝐶))
3		oveq2 7439	. . 3 ⊢ (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
4	3	eleq1d 2826	. 2 ⊢ (𝑦 = 𝑌 → ((𝑋𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑌) ∈ 𝐶))
5	2, 4	rspc2va 3634	1 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434

This theorem is referenced by:  off  7715  mgmcl  18656  submgmcl  18720  sgrppropd  18744  mndpropd  18772  issubmnd  18774  submcl  18825  issubg2  19159  gass  19319  lmodprop2d  20922  lsspropd  21016  gsummatr01lem2  22662  off2  32651  ofcf  34104  fsuppind  42600  clcllaw  48107