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Theorem caovclg 7324
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovclg.1 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)
Assertion
Ref Expression
caovclg ((𝜑 ∧ (𝐴𝐶𝐵𝐷)) → (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
StepHypRef Expression
1 caovclg.1 . . 3 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)
21ralrimivva 3159 . 2 (𝜑 → ∀𝑥𝐶𝑦𝐷 (𝑥𝐹𝑦) ∈ 𝐸)
3 oveq1 7146 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
43eleq1d 2877 . . 3 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝑦) ∈ 𝐸))
5 oveq2 7147 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
65eleq1d 2877 . . 3 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝐵) ∈ 𝐸))
74, 6rspc2v 3584 . 2 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 (𝑥𝐹𝑦) ∈ 𝐸 → (𝐴𝐹𝐵) ∈ 𝐸))
82, 7mpan9 510 1 ((𝜑 ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  (class class class)co 7139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142
This theorem is referenced by:  caovcld  7325  caovcl  7326  seqcl2  13388  seqcaopr  13407  ercpbl  16818  grprinvd  17880  gsumpropd2lem  17885  imasmnd2  17944  imasgrp2  18210  gsumzaddlem  19038  imasring  19369  qusrhm  20007  mplind  20745  plymullem  24817  fsuppssind  39452
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