Theorem brxrn 38663

Description: Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 38661, this characterizes elementhood in a range cross. (Contributed by Peter Mazsa, 27-Jun-2021.)

Assertion

Ref	Expression
brxrn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐶)))

Proof of Theorem brxrn

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

Step	Hyp	Ref	Expression
1		df-xrn 38660	. . . 4 ⊢ (𝑅 ⋉ 𝑆) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))
2	1	breqi 5106	. . 3 ⊢ (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩)
3	2	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩))
4		brin 5152	. . 3 ⊢ (𝐴((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩ ↔ (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ∧ 𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩))
5	4	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴((◡(1st ↾ (V × V)) ∘ 𝑅) ∩ (◡(2nd ↾ (V × V)) ∘ 𝑆))⟨𝐵, 𝐶⟩ ↔ (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ∧ 𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩)))
6		opex 5421	. . . . . 6 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
7		brcog 5825	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
8	6, 7	mpan2 692	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
9	8	3ad2ant1 1134	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩)))
10		brcnvg 5838	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
11	6, 10	mpan2 692	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥))
12	11	elv 3447	. . . . . . 7 ⊢ (𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥)
13		brres 5955	. . . . . . . . . . 11 ⊢ (𝑥 ∈ V → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
14	13	elv 3447	. . . . . . . . . 10 ⊢ (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥))
15		opelvvg 5675	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ⟨𝐵, 𝐶⟩ ∈ (V × V))
16	15	biantrurd 532	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩1st 𝑥)))
17	14, 16	bitr4id 290	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ ⟨𝐵, 𝐶⟩1st 𝑥))
18		br1steqg 7967	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩1st 𝑥 ↔ 𝑥 = 𝐵))
19	17, 18	bitrd 279	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ 𝑥 = 𝐵))
20	19	3adant1 1131	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(1st ↾ (V × V))𝑥 ↔ 𝑥 = 𝐵))
21	12, 20	bitrid 283	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑥 = 𝐵))
22	21	anbi1cd 636	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ 𝐴𝑅𝑥)))
23	22	exbidv 1923	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥(𝐴𝑅𝑥 ∧ 𝑥◡(1st ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴𝑅𝑥)))
24		breq2 5104	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
25	24	ceqsexgv 3610	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (∃𝑥(𝑥 = 𝐵 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
26	25	3ad2ant2 1135	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥(𝑥 = 𝐵 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐵))
27	9, 23, 26	3bitrd 305	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑅𝐵))
28		brcog 5825	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
29	6, 28	mpan2 692	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
30	29	3ad2ant1 1134	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩)))
31		brcnvg 5838	. . . . . . . . 9 ⊢ ((𝑦 ∈ V ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
32	6, 31	mpan2 692	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦))
33	32	elv 3447	. . . . . . 7 ⊢ (𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦)
34		brres 5955	. . . . . . . . . . 11 ⊢ (𝑦 ∈ V → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
35	34	elv 3447	. . . . . . . . . 10 ⊢ (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦))
36	15	biantrurd 532	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦 ↔ (⟨𝐵, 𝐶⟩ ∈ (V × V) ∧ ⟨𝐵, 𝐶⟩2nd 𝑦)))
37	35, 36	bitr4id 290	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ ⟨𝐵, 𝐶⟩2nd 𝑦))
38		br2ndeqg 7968	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩2nd 𝑦 ↔ 𝑦 = 𝐶))
39	37, 38	bitrd 279	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ 𝑦 = 𝐶))
40	39	3adant1 1131	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐵, 𝐶⟩(2nd ↾ (V × V))𝑦 ↔ 𝑦 = 𝐶))
41	33, 40	bitrid 283	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩ ↔ 𝑦 = 𝐶))
42	41	anbi1cd 636	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ (𝑦 = 𝐶 ∧ 𝐴𝑆𝑦)))
43	42	exbidv 1923	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑦(𝐴𝑆𝑦 ∧ 𝑦◡(2nd ↾ (V × V))⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑦 = 𝐶 ∧ 𝐴𝑆𝑦)))
44		breq2 5104	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝐴𝑆𝑦 ↔ 𝐴𝑆𝐶))
45	44	ceqsexgv 3610	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → (∃𝑦(𝑦 = 𝐶 ∧ 𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
46	45	3ad2ant3 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑦(𝑦 = 𝐶 ∧ 𝐴𝑆𝑦) ↔ 𝐴𝑆𝐶))
47	30, 43, 46	3bitrd 305	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩ ↔ 𝐴𝑆𝐶))
48	27, 47	anbi12d 633	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴(◡(1st ↾ (V × V)) ∘ 𝑅)⟨𝐵, 𝐶⟩ ∧ 𝐴(◡(2nd ↾ (V × V)) ∘ 𝑆)⟨𝐵, 𝐶⟩) ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐶)))
49	3, 5, 48	3bitrd 305	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902  ⟨cop 4588   class class class wbr 5100   × cxp 5632  ^◡ccnv 5633   ↾ cres 5636   ∘ ccom 5638  1^st c1st 7943  2^nd c2nd 7944   ⋉ cxrn 38454

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 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

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 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fo 6508  df-fv 6510  df-1st 7945  df-2nd 7946  df-xrn 38660

This theorem is referenced by:  brxrn2  38664  dfxrn2  38665  brxrncnvep  38666  ecxrn2  38688  brin2  38718  br1cossxrnres  38818