Theorem caovclg 7464

Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)

Hypothesis

Ref	Expression
caovclg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)

Assertion

Ref	Expression
caovclg	⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐸,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem caovclg

Step	Hyp	Ref	Expression
1		caovclg.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)
2	1	ralrimivva 3123	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ∈ 𝐸)
3		oveq1 7282	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4	3	eleq1d 2823	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝑦) ∈ 𝐸))
5		oveq2 7283	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
6	5	eleq1d 2823	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝐵) ∈ 𝐸))
7	4, 6	rspc2v 3570	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ∈ 𝐸 → (𝐴𝐹𝐵) ∈ 𝐸))
8	2, 7	mpan9 507	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  (class class class)co 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  caovcld  7465  caovcl  7466  seqcl2  13741  seqcaopr  13760  ercpbl  17260  grprinvd  18358  gsumpropd2lem  18363  imasmnd2  18422  imasgrp2  18690  gsumzaddlem  19522  imasring  19858  qusrhm  20508  mplind  21278  plymullem  25377  fsuppssind  40282