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Theorem pwmndgplus 18860

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 4153	. . 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 7733	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋 ∪ 𝑌) ∈ V)
8	2, 4, 5, 6, 7	ovmpod 7556	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 1533 ∈ wcel 2098 Vcvv 3468 ∪ cun 3941 𝒫 cpw 4597 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 Basecbs 17153 +_gcplusg 17206

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410

This theorem is referenced by: pwmndid 18861 pwmnd 18862

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