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Theorem orbsta 19353
Description: The Orbit-Stabilizer theorem. The mapping 𝐹 is a bijection from the cosets of the stabilizer subgroup of 𝐴 to the orbit of 𝐴. (Contributed by Mario Carneiro, 15-Jan-2015.)
Hypotheses
Ref Expression
gasta.1 𝑋 = (Base‘𝐺)
gasta.2 𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}
orbsta.r = (𝐺 ~QG 𝐻)
orbsta.f 𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)
orbsta.o 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
Assertion
Ref Expression
orbsta (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂)
Distinct variable groups:   𝑔,𝑘,𝑥,𝑦,   𝑢,𝑔, ,𝑘,𝑥,𝑦   𝑥,𝐻,𝑦   𝐴,𝑔,𝑘,𝑢,𝑥,𝑦   𝑔,𝐺,𝑘,𝑢,𝑥,𝑦   𝑔,𝑋,𝑘,𝑢,𝑥,𝑦   𝑘,𝑂   𝑔,𝑌,𝑘,𝑥,𝑦
Allowed substitution hints:   (𝑢)   𝐹(𝑥,𝑦,𝑢,𝑔,𝑘)   𝐻(𝑢,𝑔,𝑘)   𝑂(𝑥,𝑦,𝑢,𝑔)   𝑌(𝑢)

Proof of Theorem orbsta
Dummy variables 𝑎 𝑏 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gasta.1 . . . . 5 𝑋 = (Base‘𝐺)
2 gasta.2 . . . . 5 𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}
3 orbsta.r . . . . 5 = (𝐺 ~QG 𝐻)
4 orbsta.f . . . . 5 𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)
51, 2, 3, 4orbstafun 19351 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Fun 𝐹)
6 simpr 484 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐴𝑌)
76adantr 480 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝐴𝑌)
81gaf 19335 . . . . . . . . . 10 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
98adantr 480 . . . . . . . . 9 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → :(𝑋 × 𝑌)⟶𝑌)
109adantr 480 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → :(𝑋 × 𝑌)⟶𝑌)
11 simpr 484 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝑘𝑋)
1210, 11, 7fovcdmd 7622 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → (𝑘 𝐴) ∈ 𝑌)
13 eqid 2740 . . . . . . . 8 (𝑘 𝐴) = (𝑘 𝐴)
14 oveq1 7455 . . . . . . . . . 10 ( = 𝑘 → ( 𝐴) = (𝑘 𝐴))
1514eqeq1d 2742 . . . . . . . . 9 ( = 𝑘 → (( 𝐴) = (𝑘 𝐴) ↔ (𝑘 𝐴) = (𝑘 𝐴)))
1615rspcev 3635 . . . . . . . 8 ((𝑘𝑋 ∧ (𝑘 𝐴) = (𝑘 𝐴)) → ∃𝑋 ( 𝐴) = (𝑘 𝐴))
1711, 13, 16sylancl 585 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → ∃𝑋 ( 𝐴) = (𝑘 𝐴))
18 orbsta.o . . . . . . . 8 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
1918gaorb 19347 . . . . . . 7 (𝐴𝑂(𝑘 𝐴) ↔ (𝐴𝑌 ∧ (𝑘 𝐴) ∈ 𝑌 ∧ ∃𝑋 ( 𝐴) = (𝑘 𝐴)))
207, 12, 17, 19syl3anbrc 1343 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → 𝐴𝑂(𝑘 𝐴))
21 ovex 7481 . . . . . . 7 (𝑘 𝐴) ∈ V
22 elecg 8807 . . . . . . 7 (((𝑘 𝐴) ∈ V ∧ 𝐴𝑌) → ((𝑘 𝐴) ∈ [𝐴]𝑂𝐴𝑂(𝑘 𝐴)))
2321, 7, 22sylancr 586 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → ((𝑘 𝐴) ∈ [𝐴]𝑂𝐴𝑂(𝑘 𝐴)))
2420, 23mpbird 257 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑘𝑋) → (𝑘 𝐴) ∈ [𝐴]𝑂)
251, 2gastacl 19349 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
261, 3eqger 19218 . . . . . 6 (𝐻 ∈ (SubGrp‘𝐺) → Er 𝑋)
2725, 26syl 17 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Er 𝑋)
281fvexi 6934 . . . . . 6 𝑋 ∈ V
2928a1i 11 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝑋 ∈ V)
304, 24, 27, 29qliftf 8863 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → (Fun 𝐹𝐹:(𝑋 / )⟶[𝐴]𝑂))
315, 30mpbid 232 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )⟶[𝐴]𝑂)
32 eqid 2740 . . . . 5 (𝑋 / ) = (𝑋 / )
33 fveqeq2 6929 . . . . . . 7 ([𝑧] = 𝑎 → ((𝐹‘[𝑧] ) = (𝐹𝑏) ↔ (𝐹𝑎) = (𝐹𝑏)))
34 eqeq1 2744 . . . . . . 7 ([𝑧] = 𝑎 → ([𝑧] = 𝑏𝑎 = 𝑏))
3533, 34imbi12d 344 . . . . . 6 ([𝑧] = 𝑎 → (((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏) ↔ ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
3635ralbidv 3184 . . . . 5 ([𝑧] = 𝑎 → (∀𝑏 ∈ (𝑋 / )((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏) ↔ ∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
37 fveq2 6920 . . . . . . . . 9 ([𝑤] = 𝑏 → (𝐹‘[𝑤] ) = (𝐹𝑏))
3837eqeq2d 2751 . . . . . . . 8 ([𝑤] = 𝑏 → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ (𝐹‘[𝑧] ) = (𝐹𝑏)))
39 eqeq2 2752 . . . . . . . 8 ([𝑤] = 𝑏 → ([𝑧] = [𝑤] ↔ [𝑧] = 𝑏))
4038, 39imbi12d 344 . . . . . . 7 ([𝑤] = 𝑏 → (((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ) ↔ ((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏)))
411, 2, 3, 4orbstaval 19352 . . . . . . . . . . . 12 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) → (𝐹‘[𝑧] ) = (𝑧 𝐴))
4241adantrr 716 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹‘[𝑧] ) = (𝑧 𝐴))
431, 2, 3, 4orbstaval 19352 . . . . . . . . . . . 12 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → (𝐹‘[𝑤] ) = (𝑤 𝐴))
4443adantrl 715 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝐹‘[𝑤] ) = (𝑤 𝐴))
4542, 44eqeq12d 2756 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ (𝑧 𝐴) = (𝑤 𝐴)))
461, 2, 3gastacos 19350 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝑧 𝑤 ↔ (𝑧 𝐴) = (𝑤 𝐴)))
4727adantr 480 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → Er 𝑋)
48 simprl 770 . . . . . . . . . . 11 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → 𝑧𝑋)
4947, 48erth 8814 . . . . . . . . . 10 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → (𝑧 𝑤 ↔ [𝑧] = [𝑤] ))
5045, 46, 493bitr2d 307 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) ↔ [𝑧] = [𝑤] ))
5150biimpd 229 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ))
5251anassrs 467 . . . . . . 7 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) ∧ 𝑤𝑋) → ((𝐹‘[𝑧] ) = (𝐹‘[𝑤] ) → [𝑧] = [𝑤] ))
5332, 40, 52ectocld 8842 . . . . . 6 (((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) ∧ 𝑏 ∈ (𝑋 / )) → ((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏))
5453ralrimiva 3152 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑧𝑋) → ∀𝑏 ∈ (𝑋 / )((𝐹‘[𝑧] ) = (𝐹𝑏) → [𝑧] = 𝑏))
5532, 36, 54ectocld 8842 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑎 ∈ (𝑋 / )) → ∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
5655ralrimiva 3152 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ∀𝑎 ∈ (𝑋 / )∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
57 dff13 7292 . . 3 (𝐹:(𝑋 / )–1-1→[𝐴]𝑂 ↔ (𝐹:(𝑋 / )⟶[𝐴]𝑂 ∧ ∀𝑎 ∈ (𝑋 / )∀𝑏 ∈ (𝑋 / )((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
5831, 56, 57sylanbrc 582 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1→[𝐴]𝑂)
59 vex 3492 . . . . . . . . 9 ∈ V
60 elecg 8807 . . . . . . . . 9 (( ∈ V ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂𝐴𝑂))
6159, 6, 60sylancr 586 . . . . . . . 8 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂𝐴𝑂))
6218gaorb 19347 . . . . . . . 8 (𝐴𝑂 ↔ (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = ))
6361, 62bitrdi 287 . . . . . . 7 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( ∈ [𝐴]𝑂 ↔ (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = )))
6463biimpa 476 . . . . . 6 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → (𝐴𝑌𝑌 ∧ ∃𝑤𝑋 (𝑤 𝐴) = ))
6564simp3d 1144 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → ∃𝑤𝑋 (𝑤 𝐴) = )
663ovexi 7482 . . . . . . . . . 10 ∈ V
6766ecelqsi 8831 . . . . . . . . 9 (𝑤𝑋 → [𝑤] ∈ (𝑋 / ))
6843eqcomd 2746 . . . . . . . . 9 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → (𝑤 𝐴) = (𝐹‘[𝑤] ))
69 fveq2 6920 . . . . . . . . . 10 (𝑧 = [𝑤] → (𝐹𝑧) = (𝐹‘[𝑤] ))
7069rspceeqv 3658 . . . . . . . . 9 (([𝑤] ∈ (𝑋 / ) ∧ (𝑤 𝐴) = (𝐹‘[𝑤] )) → ∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧))
7167, 68, 70syl2an2 685 . . . . . . . 8 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → ∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧))
72 eqeq1 2744 . . . . . . . . 9 ((𝑤 𝐴) = → ((𝑤 𝐴) = (𝐹𝑧) ↔ = (𝐹𝑧)))
7372rexbidv 3185 . . . . . . . 8 ((𝑤 𝐴) = → (∃𝑧 ∈ (𝑋 / )(𝑤 𝐴) = (𝐹𝑧) ↔ ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
7471, 73syl5ibcom 245 . . . . . . 7 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑤𝑋) → ((𝑤 𝐴) = → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
7574rexlimdva 3161 . . . . . 6 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → (∃𝑤𝑋 (𝑤 𝐴) = → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
7675imp 406 . . . . 5 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∃𝑤𝑋 (𝑤 𝐴) = ) → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧))
7765, 76syldan 590 . . . 4 ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ ∈ [𝐴]𝑂) → ∃𝑧 ∈ (𝑋 / ) = (𝐹𝑧))
7877ralrimiva 3152 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ∀ ∈ [ 𝐴]𝑂𝑧 ∈ (𝑋 / ) = (𝐹𝑧))
79 dffo3 7136 . . 3 (𝐹:(𝑋 / )–onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / )⟶[𝐴]𝑂 ∧ ∀ ∈ [ 𝐴]𝑂𝑧 ∈ (𝑋 / ) = (𝐹𝑧)))
8031, 78, 79sylanbrc 582 . 2 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–onto→[𝐴]𝑂)
81 df-f1o 6580 . 2 (𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / )–1-1→[𝐴]𝑂𝐹:(𝑋 / )–onto→[𝐴]𝑂))
8258, 80, 81sylanbrc 582 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  wral 3067  wrex 3076  {crab 3443  Vcvv 3488  wss 3976  {cpr 4650  cop 4654   class class class wbr 5166  {copab 5228  cmpt 5249   × cxp 5698  ran crn 5701  Fun wfun 6567  wf 6569  1-1wf1 6570  ontowfo 6571  1-1-ontowf1o 6572  cfv 6573  (class class class)co 7448   Er wer 8760  [cec 8761   / cqs 8762  Basecbs 17258  SubGrpcsubg 19160   ~QG cqg 19162   GrpAct cga 19329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-ec 8765  df-qs 8769  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-subg 19163  df-eqg 19165  df-ga 19330
This theorem is referenced by:  orbsta2  19354
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