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Theorem pwmndgplus 18746
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 4119 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝑦) = (𝑋𝑌))
43adantl 483 . 2 (((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝑦) = (𝑋𝑌))
5 simpl 484 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
6 simpr 486 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → 𝑌 ∈ 𝒫 𝐴)
7 unexg 7684 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (𝑋𝑌) ∈ V)
82, 4, 5, 6, 7ovmpod 7508 1 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (𝑋(+g𝑀)𝑌) = (𝑋𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3446  cun 3909  𝒫 cpw 4561  cfv 6497  (class class class)co 7358  cmpo 7360  Basecbs 17084  +gcplusg 17134
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  pwmndid  18747  pwmnd  18748
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