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Theorem br1cossxrnres 35219
Description: 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.)
Assertion
Ref Expression
br1cossxrnres (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝐷   𝑢,𝐸   𝑢,𝑅   𝑢,𝑆   𝑢,𝑉   𝑢,𝑊   𝑢,𝑋   𝑢,𝑌

Proof of Theorem br1cossxrnres
StepHypRef Expression
1 xrnres2 35182 . . . 4 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
21cosseqi 35203 . . 3 ≀ ((𝑅𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆𝐴))
32breqi 4968 . 2 (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩)
4 opex 5248 . . . 4 𝐵, 𝐶⟩ ∈ V
5 opex 5248 . . . 4 𝐷, 𝐸⟩ ∈ V
6 br1cossres 35215 . . . 4 ((⟨𝐵, 𝐶⟩ ∈ V ∧ ⟨𝐷, 𝐸⟩ ∈ V) → (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩)))
74, 5, 6mp2an 688 . . 3 (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩))
8 brxrn 35157 . . . . . . 7 ((𝑢 ∈ V ∧ 𝐵𝑉𝐶𝑊) → (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
98el3v1 35024 . . . . . 6 ((𝐵𝑉𝐶𝑊) → (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵𝑢𝑆𝐶)))
10 brxrn 35157 . . . . . . 7 ((𝑢 ∈ V ∧ 𝐷𝑋𝐸𝑌) → (𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷𝑢𝑆𝐸)))
1110el3v1 35024 . . . . . 6 ((𝐷𝑋𝐸𝑌) → (𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷𝑢𝑆𝐸)))
129, 11bi2anan9 635 . . . . 5 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → ((𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩) ↔ ((𝑢𝑅𝐵𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷𝑢𝑆𝐸))))
13 an2anr 35030 . . . . 5 (((𝑢𝑅𝐵𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷)))
1412, 13syl6bb 288 . . . 4 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → ((𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩) ↔ ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
1514rexbidv 3260 . . 3 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (∃𝑢𝐴 (𝑢(𝑅𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅𝑆)⟨𝐷, 𝐸⟩) ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
167, 15syl5bb 284 . 2 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ ((𝑅𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
173, 16syl5bbr 286 1 (((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2081  wrex 3106  Vcvv 3437  cop 4478   class class class wbr 4962  cres 5445  cxrn 34984  ccoss 34985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fo 6231  df-fv 6233  df-1st 7545  df-2nd 7546  df-xrn 35154  df-coss 35190
This theorem is referenced by:  br1cossxrnidres  35222  br1cossxrncnvepres  35223  br1cossxrncnvssrres  35279
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