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Theorem pwmndgplus 18970

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	⊢ (+_g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))

**Assertion**
Ref	Expression
pwmndgplus	⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋(+_g‘𝑀)𝑌) = (𝑋 ∪ 𝑌))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝑋,𝑦 𝑥,𝑌,𝑦 Allowed substitution hints: 𝑀(𝑥,𝑦)

**Proof of Theorem pwmndgplus**
Step	Hyp	Ref	Expression
1		pwmnd.p	. . 3 ⊢ (+_g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))
2	1	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (+_g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦)))
3		uneq12 4176	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ∪ 𝑦) = (𝑋 ∪ 𝑌))
4	3	adantl 481	. 2 ⊢ (((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥 ∪ 𝑦) = (𝑋 ∪ 𝑌))
5		simpl 482	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
6		simpr 484	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → 𝑌 ∈ 𝒫 𝐴)
7		unexg 7769	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋 ∪ 𝑌) ∈ V)
8	2, 4, 5, 6, 7	ovmpod 7592	1 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋(+_g‘𝑀)𝑌) = (𝑋 ∪ 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3481 ∪ cun 3964 𝒫 cpw 4608 ‘cfv 6569 (class class class)co 7438 ∈ cmpo 7440 Basecbs 17254 +_gcplusg 17307

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5305 ax-nul 5315 ax-pr 5441 ax-un 7761

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3483 df-sbc 3795 df-dif 3969 df-un 3971 df-ss 3983 df-nul 4343 df-if 4535 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-br 5152 df-opab 5214 df-id 5587 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-iota 6522 df-fun 6571 df-fv 6577 df-ov 7441 df-oprab 7442 df-mpo 7443

This theorem is referenced by: pwmndid 18971 pwmnd 18972

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