Theorem caovclg 7602

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 3208	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ∈ 𝐸)
3		oveq1 7417	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4	3	eleq1d 2848	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝑦) ∈ 𝐸))
5		oveq2 7418	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
6	5	eleq1d 2848	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝐵) ∈ 𝐸))
7	4, 6	rspc2v 3592	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ∈ 𝐸 → (𝐴𝐹𝐵) ∈ 𝐸))
8	2, 7	mpan9 515	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is referenced by:  caovcld  7603  caovcl  7604  coof  7698  seqcl2  14052  seqcaopr  14071  ercpbl  17598  grpinva  18727  gsumpropd2lem  18732  imasmnd2  18827  imasgrp2  19116  gsumzaddlem  19986  imasrng  20250  imasring  20408  qusrhm  21415  qusmul2idl  21418  mplind  22221  plymullem  26373  fsuppssind  43325