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Theorem pwmndgplus 18998

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 4118	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ∪ 𝑦) = (𝑋 ∪ 𝑌))
4	3	adantl 486	. 2 ⊢ (((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥 ∪ 𝑦) = (𝑋 ∪ 𝑌))
5		simpl 487	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
6		simpr 489	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → 𝑌 ∈ 𝒫 𝐴)
7		unexg 7743	. 2 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋 ∪ 𝑌) ∈ V)
8	2, 4, 5, 6, 7	ovmpod 7564	1 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋(+_g‘𝑀)𝑌) = (𝑋 ∪ 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ∪ cun 3904 𝒫 cpw 4563 ‘cfv 6538 (class class class)co 7412 ∈ cmpo 7414 Basecbs 17270 +_gcplusg 17311

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-iota 6494 df-fun 6540 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417

This theorem is referenced by: pwmndid 18999 pwmnd 19000

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