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Theorem pwmndgplus 18868

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 4134	. . 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 7726	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋 ∪ 𝑌) ∈ V)
8	2, 4, 5, 6, 7	ovmpod 7548	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 1540 ∈ wcel 2109 Vcvv 3455 ∪ cun 3920 𝒫 cpw 4571 ‘cfv 6519 (class class class)co 7394 ∈ cmpo 7396 Basecbs 17185 +_gcplusg 17226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5259 ax-nul 5269 ax-pr 5395 ax-un 7718

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ral 3047 df-rex 3056 df-rab 3412 df-v 3457 df-sbc 3762 df-dif 3925 df-un 3927 df-ss 3939 df-nul 4305 df-if 4497 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-br 5116 df-opab 5178 df-id 5541 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-iota 6472 df-fun 6521 df-fv 6527 df-ov 7397 df-oprab 7398 df-mpo 7399

This theorem is referenced by: pwmndid 18869 pwmnd 18870

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