Theorem ecxrn2 38659

Description: The (𝑅 ⋉ 𝑆)-coset of a set is the Cartesian product of its 𝑅-coset and 𝑆-coset. (Contributed by Peter Mazsa, 16-Oct-2020.)

Assertion

Ref	Expression
ecxrn2	⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = ([𝐴]𝑅 × [𝐴]𝑆))

Proof of Theorem ecxrn2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relecxrn 38658	. . 3 ⊢ (𝐴 ∈ 𝑉 → Rel [𝐴](𝑅 ⋉ 𝑆))
2		relxp 5650	. . 3 ⊢ Rel ([𝐴]𝑅 × [𝐴]𝑆)
3	1, 2	jctir 520	. 2 ⊢ (𝐴 ∈ 𝑉 → (Rel [𝐴](𝑅 ⋉ 𝑆) ∧ Rel ([𝐴]𝑅 × [𝐴]𝑆)))
4		brxrn 38634	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
5	4	el3v23 38485	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
6		opex 5419	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
7		elecALTV 38522	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⟨𝑥, 𝑦⟩ ∈ V) → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
8	6, 7	mpan2 692	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
9		elecALTV 38522	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
10	9	elvd 3448	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
11		elecALTV 38522	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑆 ↔ 𝐴𝑆𝑦))
12	11	elvd 3448	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]𝑆 ↔ 𝐴𝑆𝑦))
13	10, 12	anbi12d 633	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ [𝐴]𝑅 ∧ 𝑦 ∈ [𝐴]𝑆) ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
14	5, 8, 13	3bitr4d 311	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ (𝑥 ∈ [𝐴]𝑅 ∧ 𝑦 ∈ [𝐴]𝑆)))
15		opelxp 5668	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ([𝐴]𝑅 × [𝐴]𝑆) ↔ (𝑥 ∈ [𝐴]𝑅 ∧ 𝑦 ∈ [𝐴]𝑆))
16	14, 15	bitr4di 289	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ⟨𝑥, 𝑦⟩ ∈ ([𝐴]𝑅 × [𝐴]𝑆)))
17	16	eqrelrdv2 5752	. 2 ⊢ (((Rel [𝐴](𝑅 ⋉ 𝑆) ∧ Rel ([𝐴]𝑅 × [𝐴]𝑆)) ∧ 𝐴 ∈ 𝑉) → [𝐴](𝑅 ⋉ 𝑆) = ([𝐴]𝑅 × [𝐴]𝑆))
18	3, 17	mpancom 689	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = ([𝐴]𝑅 × [𝐴]𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588   class class class wbr 5100   × cxp 5630  Rel wrel 5637  [cec 8643   ⋉ cxrn 38425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-1st 7943  df-2nd 7944  df-ec 8647  df-xrn 38631

This theorem is referenced by:  ecxrncnvep2  38661