Theorem ecxrn2 38532

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 38531	. . 3 ⊢ (𝐴 ∈ 𝑉 → Rel [𝐴](𝑅 ⋉ 𝑆))
2		relxp 5640	. . 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 38507	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
5	4	el3v23 38369	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
6		opex 5410	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
7		elecALTV 38403	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⟨𝑥, 𝑦⟩ ∈ V) → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
8	6, 7	mpan2 691	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
9		elecALTV 38403	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
10	9	elvd 3444	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
11		elecALTV 38403	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑆 ↔ 𝐴𝑆𝑦))
12	11	elvd 3444	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]𝑆 ↔ 𝐴𝑆𝑦))
13	10, 12	anbi12d 632	. . . . 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 5658	. . . 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 5742	. 2 ⊢ (((Rel [𝐴](𝑅 ⋉ 𝑆) ∧ Rel ([𝐴]𝑅 × [𝐴]𝑆)) ∧ 𝐴 ∈ 𝑉) → [𝐴](𝑅 ⋉ 𝑆) = ([𝐴]𝑅 × [𝐴]𝑆))
18	3, 17	mpancom 688	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = ([𝐴]𝑅 × [𝐴]𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ⟨cop 4584   class class class wbr 5096   × cxp 5620  Rel wrel 5627  [cec 8631   ⋉ cxrn 38314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-1st 7931  df-2nd 7932  df-ec 8635  df-xrn 38504

This theorem is referenced by:  ecxrncnvep2  38534