Theorem ecxrn2 38775

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 38774	. . 3 ⊢ (𝐴 ∈ 𝑉 → Rel [𝐴](𝑅 ⋉ 𝑆))
2		relxp 5636	. . 3 ⊢ Rel ([𝐴]𝑅 × [𝐴]𝑆)
3	1, 2	jctir 525	. 2 ⊢ (𝐴 ∈ 𝑉 → (Rel [𝐴](𝑅 ⋉ 𝑆) ∧ Rel ([𝐴]𝑅 × [𝐴]𝑆)))
4		brxrn 38750	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
5	4	el3v23 38601	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
6		opex 5403	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
7		elecALTV 38638	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⟨𝑥, 𝑦⟩ ∈ V) → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
8	6, 7	mpan2 697	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
9		elecALTV 38638	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
10	9	elvd 3437	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
11		elecALTV 38638	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑆 ↔ 𝐴𝑆𝑦))
12	11	elvd 3437	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]𝑆 ↔ 𝐴𝑆𝑦))
13	10, 12	anbi12d 638	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ [𝐴]𝑅 ∧ 𝑦 ∈ [𝐴]𝑆) ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
14	5, 8, 13	3bitr4d 312	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ (𝑥 ∈ [𝐴]𝑅 ∧ 𝑦 ∈ [𝐴]𝑆)))
15		opelxp 5654	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ([𝐴]𝑅 × [𝐴]𝑆) ↔ (𝑥 ∈ [𝐴]𝑅 ∧ 𝑦 ∈ [𝐴]𝑆))
16	14, 15	bitr4di 290	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ⟨𝑥, 𝑦⟩ ∈ ([𝐴]𝑅 × [𝐴]𝑆)))
17	16	eqrelrdv2 5738	. 2 ⊢ (((Rel [𝐴](𝑅 ⋉ 𝑆) ∧ Rel ([𝐴]𝑅 × [𝐴]𝑆)) ∧ 𝐴 ∈ 𝑉) → [𝐴](𝑅 ⋉ 𝑆) = ([𝐴]𝑅 × [𝐴]𝑆))
18	3, 17	mpancom 694	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = ([𝐴]𝑅 × [𝐴]𝑆))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561   class class class wbr 5072   × cxp 5616  Rel wrel 5623  [cec 8631   ⋉ cxrn 38541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493  df-1st 7931  df-2nd 7932  df-ec 8635  df-xrn 38747

This theorem is referenced by:  ecxrncnvep2  38777