Theorem ovrspc2v 6905

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 6886	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐹𝑦) = (𝑋𝐹𝑦))
2	1	eleq1d 2864	. 2 ⊢ (𝑥 = 𝑋 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑦) ∈ 𝐶))
3		oveq2 6887	. . 3 ⊢ (𝑦 = 𝑌 → (𝑋𝐹𝑦) = (𝑋𝐹𝑌))
4	3	eleq1d 2864	. 2 ⊢ (𝑦 = 𝑌 → ((𝑋𝐹𝑦) ∈ 𝐶 ↔ (𝑋𝐹𝑌) ∈ 𝐶))
5	2, 4	rspc2va 3512	1 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) → (𝑋𝐹𝑌) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∀wral 3090  (class class class)co 6879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-iota 6065  df-fv 6110  df-ov 6882

This theorem is referenced by:  mgmcl  17559  mndpropd  17630  issubmnd  17632  submcl  17667  issubg2  17921  gass  18045  lmodprop2d  19242  lsspropd  19337  gsummatr01lem2  20788  off2  29961  ofcf  30680  submgmcl  42588  clcllaw  42621