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Theorem releqgg 13011
Description: The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypothesis
Ref Expression
releqg.r 𝑅 = (𝐺 ~QG 𝑆)
Assertion
Ref Expression
releqgg ((𝐺𝑉𝑆𝑊) → Rel 𝑅)

Proof of Theorem releqgg
Dummy variables 𝑖 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4752 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)}
2 releqg.r . . . 4 𝑅 = (𝐺 ~QG 𝑆)
3 elex 2748 . . . . . 6 (𝐺𝑉𝐺 ∈ V)
43adantr 276 . . . . 5 ((𝐺𝑉𝑆𝑊) → 𝐺 ∈ V)
5 elex 2748 . . . . . 6 (𝑆𝑊𝑆 ∈ V)
65adantl 277 . . . . 5 ((𝐺𝑉𝑆𝑊) → 𝑆 ∈ V)
7 vex 2740 . . . . . . . . 9 𝑥 ∈ V
8 vex 2740 . . . . . . . . 9 𝑦 ∈ V
97, 8prss 3748 . . . . . . . 8 ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐺))
109anbi1i 458 . . . . . . 7 (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆) ↔ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆))
1110opabbii 4069 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)}
12 basfn 12512 . . . . . . . . 9 Base Fn V
13 funfvex 5531 . . . . . . . . . 10 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
1413funfni 5315 . . . . . . . . 9 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
1512, 4, 14sylancr 414 . . . . . . . 8 ((𝐺𝑉𝑆𝑊) → (Base‘𝐺) ∈ V)
16 xpexg 4739 . . . . . . . 8 (((Base‘𝐺) ∈ V ∧ (Base‘𝐺) ∈ V) → ((Base‘𝐺) × (Base‘𝐺)) ∈ V)
1715, 15, 16syl2anc 411 . . . . . . 7 ((𝐺𝑉𝑆𝑊) → ((Base‘𝐺) × (Base‘𝐺)) ∈ V)
18 opabssxp 4699 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)} ⊆ ((Base‘𝐺) × (Base‘𝐺))
1918a1i 9 . . . . . . 7 ((𝐺𝑉𝑆𝑊) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)} ⊆ ((Base‘𝐺) × (Base‘𝐺)))
2017, 19ssexd 4142 . . . . . 6 ((𝐺𝑉𝑆𝑊) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)} ∈ V)
2111, 20eqeltrrid 2265 . . . . 5 ((𝐺𝑉𝑆𝑊) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)} ∈ V)
22 fveq2 5514 . . . . . . . . 9 (𝑟 = 𝐺 → (Base‘𝑟) = (Base‘𝐺))
2322sseq2d 3185 . . . . . . . 8 (𝑟 = 𝐺 → ({𝑥, 𝑦} ⊆ (Base‘𝑟) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐺)))
24 fveq2 5514 . . . . . . . . . 10 (𝑟 = 𝐺 → (+g𝑟) = (+g𝐺))
25 fveq2 5514 . . . . . . . . . . 11 (𝑟 = 𝐺 → (invg𝑟) = (invg𝐺))
2625fveq1d 5516 . . . . . . . . . 10 (𝑟 = 𝐺 → ((invg𝑟)‘𝑥) = ((invg𝐺)‘𝑥))
27 eqidd 2178 . . . . . . . . . 10 (𝑟 = 𝐺𝑦 = 𝑦)
2824, 26, 27oveq123d 5893 . . . . . . . . 9 (𝑟 = 𝐺 → (((invg𝑟)‘𝑥)(+g𝑟)𝑦) = (((invg𝐺)‘𝑥)(+g𝐺)𝑦))
2928eleq1d 2246 . . . . . . . 8 (𝑟 = 𝐺 → ((((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖 ↔ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑖))
3023, 29anbi12d 473 . . . . . . 7 (𝑟 = 𝐺 → (({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖) ↔ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑖)))
3130opabbidv 4068 . . . . . 6 (𝑟 = 𝐺 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑖)})
32 eleq2 2241 . . . . . . . 8 (𝑖 = 𝑆 → ((((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑖 ↔ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆))
3332anbi2d 464 . . . . . . 7 (𝑖 = 𝑆 → (({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑖) ↔ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)))
3433opabbidv 4068 . . . . . 6 (𝑖 = 𝑆 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑖)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)})
35 df-eqg 12963 . . . . . 6 ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg𝑟)‘𝑥)(+g𝑟)𝑦) ∈ 𝑖)})
3631, 34, 35ovmpog 6006 . . . . 5 ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)} ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)})
374, 6, 21, 36syl3anc 1238 . . . 4 ((𝐺𝑉𝑆𝑊) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)})
382, 37eqtrid 2222 . . 3 ((𝐺𝑉𝑆𝑊) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)})
3938releqd 4709 . 2 ((𝐺𝑉𝑆𝑊) → (Rel 𝑅 ↔ Rel {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝐺) ∧ (((invg𝐺)‘𝑥)(+g𝐺)𝑦) ∈ 𝑆)}))
401, 39mpbiri 168 1 ((𝐺𝑉𝑆𝑊) → Rel 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  wss 3129  {cpr 3593  {copab 4062   × cxp 4623  Rel wrel 4630   Fn wfn 5210  cfv 5215  (class class class)co 5872  Basecbs 12454  +gcplusg 12528  invgcminusg 12810   ~QG cqg 12960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-inn 8916  df-ndx 12457  df-slot 12458  df-base 12460  df-eqg 12963
This theorem is referenced by:  eqger  13014  eqgid  13016
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