MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwmndgplus Structured version   Visualization version   GIF version

Theorem pwmndgplus 18889
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 4151 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝑦) = (𝑋𝑌))
43adantl 480 . 2 (((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝑦) = (𝑋𝑌))
5 simpl 481 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
6 simpr 483 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → 𝑌 ∈ 𝒫 𝐴)
7 unexg 7748 . 2 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (𝑋𝑌) ∈ V)
82, 4, 5, 6, 7ovmpod 7569 1 ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (𝑋(+g𝑀)𝑌) = (𝑋𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3463  cun 3938  𝒫 cpw 4598  cfv 6542  (class class class)co 7415  cmpo 7417  Basecbs 17177  +gcplusg 17230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420
This theorem is referenced by:  pwmndid  18890  pwmnd  18891
  Copyright terms: Public domain W3C validator