Theorem ecxrn2 39157

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 39156	. . 3 ⊢ (𝐴 ∈ 𝑉 → Rel [𝐴](𝑅 ⋉ 𝑆))
2		relxp 5673	. . 3 ⊢ Rel ([𝐴]𝑅 × [𝐴]𝑆)
3	1, 2	jctir 530	. 2 ⊢ (𝐴 ∈ 𝑉 → (Rel [𝐴](𝑅 ⋉ 𝑆) ∧ Rel ([𝐴]𝑅 × [𝐴]𝑆)))
4		brxrn 39132	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
5	4	el3v23 38983	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
6		opex 5439	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
7		elecALTV 39020	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⟨𝑥, 𝑦⟩ ∈ V) → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
8	6, 7	mpan2 704	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝑥, 𝑦⟩))
9		elecALTV 39020	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
10	9	elvd 3456	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
11		elecALTV 39020	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝐴]𝑆 ↔ 𝐴𝑆𝑦))
12	11	elvd 3456	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]𝑆 ↔ 𝐴𝑆𝑦))
13	10, 12	anbi12d 644	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ [𝐴]𝑅 ∧ 𝑦 ∈ [𝐴]𝑆) ↔ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
14	5, 8, 13	3bitr4d 314	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ (𝑥 ∈ [𝐴]𝑅 ∧ 𝑦 ∈ [𝐴]𝑆)))
15		opelxp 5691	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ([𝐴]𝑅 × [𝐴]𝑆) ↔ (𝑥 ∈ [𝐴]𝑅 ∧ 𝑦 ∈ [𝐴]𝑆))
16	14, 15	bitr4di 292	. . 3 ⊢ (𝐴 ∈ 𝑉 → (⟨𝑥, 𝑦⟩ ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ⟨𝑥, 𝑦⟩ ∈ ([𝐴]𝑅 × [𝐴]𝑆)))
17	16	eqrelrdv2 5775	. 2 ⊢ (((Rel [𝐴](𝑅 ⋉ 𝑆) ∧ Rel ([𝐴]𝑅 × [𝐴]𝑆)) ∧ 𝐴 ∈ 𝑉) → [𝐴](𝑅 ⋉ 𝑆) = ([𝐴]𝑅 × [𝐴]𝑆))
18	3, 17	mpancom 701	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = ([𝐴]𝑅 × [𝐴]𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⟨cop 4590   class class class wbr 5103   × cxp 5653  Rel wrel 5660  [cec 8695   ⋉ cxrn 38923

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-1st 7987  df-2nd 7988  df-ec 8699  df-xrn 39129

This theorem is used by:  ecxrncnvep2  39159