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Theorem caovclg 5680
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 2418 . 2 (𝜑 → ∀𝑥𝐶𝑦𝐷 (𝑥𝐹𝑦) ∈ 𝐸)
3 oveq1 5546 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
43eleq1d 2122 . . 3 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝑦) ∈ 𝐸))
5 oveq2 5547 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
65eleq1d 2122 . . 3 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝐵) ∈ 𝐸))
74, 6rspc2v 2684 . 2 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 (𝑥𝐹𝑦) ∈ 𝐸 → (𝐴𝐹𝐵) ∈ 𝐸))
82, 7mpan9 269 1 ((𝜑 ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  wral 2323  (class class class)co 5539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937  df-ov 5542
This theorem is referenced by:  caovcld  5681  caovcl  5682  caovlem2d  5720  grprinvd  5723  frec2uzrdg  9353  frecuzrdgsuc  9359  iseqovex  9377  iseqval  9378  iseqcaopr  9400
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