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Theorem pwmndgplus 18858
Description: The operation of the monoid of the power set of a class 𝐴 under union. (Contributed by AV, 27-Feb-2024.)
Hypotheses
Ref Expression
pwmnd.b (Base‘𝑀) = 𝒫 𝐴
pwmnd.p (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))
Assertion
Ref Expression
pwmndgplus ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (𝑋(+g𝑀)𝑌) = (𝑋𝑌))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝑀(𝑥,𝑦)

Proof of Theorem pwmndgplus
StepHypRef Expression
1 pwmnd.p . . 3 (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))
21a1i 11 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦)))
3 uneq12 4158 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝑦) = (𝑋𝑌))
43adantl 481 . 2 (((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝑦) = (𝑋𝑌))
5 simpl 482 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
6 simpr 484 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → 𝑌 ∈ 𝒫 𝐴)
7 unexg 7740 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (𝑋𝑌) ∈ V)
82, 4, 5, 6, 7ovmpod 7563 1 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (𝑋(+g𝑀)𝑌) = (𝑋𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  cun 3946  𝒫 cpw 4602  cfv 6543  (class class class)co 7412  cmpo 7414  Basecbs 17151  +gcplusg 17204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417
This theorem is referenced by:  pwmndid  18859  pwmnd  18860
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