Theorem orbsta 19254

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 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}
orbsta.r	⊢ ∼ = (𝐺 ~QG 𝐻)
orbsta.f	⊢ 𝐹 = ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩)
orbsta.o	⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}

Assertion

Ref	Expression
orbsta	⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 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 ℎ 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gasta.1	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
2		gasta.2	. . . . 5 ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}
3		orbsta.r	. . . . 5 ⊢ ∼ = (𝐺 ~QG 𝐻)
4		orbsta.f	. . . . 5 ⊢ 𝐹 = ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩)
5	1, 2, 3, 4	orbstafun 19252	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → Fun 𝐹)
6		simpr 484	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ 𝑌)
7	6	adantr 480	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ 𝑌)
8	1	gaf 19236	. . . . . . . . . 10 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
9	8	adantr 480	. . . . . . . . 9 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
10	9	adantr 480	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
11		simpr 484	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
12	10, 11, 7	fovcdmd 7540	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ 𝑌)
13		eqid 2737	. . . . . . . 8 ⊢ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)
14		oveq1 7375	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
15	14	eqeq1d 2739	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴) ↔ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
16	15	rspcev 3578	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝑋 ∧ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
17	11, 13, 16	sylancl 587	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
18		orbsta.o	. . . . . . . 8 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
19	18	gaorb 19248	. . . . . . 7 ⊢ (𝐴𝑂(𝑘 ⊕ 𝐴) ↔ (𝐴 ∈ 𝑌 ∧ (𝑘 ⊕ 𝐴) ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
20	7, 12, 17, 19	syl3anbrc 1345	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝐴𝑂(𝑘 ⊕ 𝐴))
21		ovex 7401	. . . . . . 7 ⊢ (𝑘 ⊕ 𝐴) ∈ V
22		elecg 8690	. . . . . . 7 ⊢ (((𝑘 ⊕ 𝐴) ∈ V ∧ 𝐴 ∈ 𝑌) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
23	21, 7, 22	sylancr 588	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
24	20, 23	mpbird 257	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂)
25	1, 2	gastacl 19250	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
26	1, 3	eqger 19119	. . . . . 6 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
27	25, 26	syl 17	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∼ Er 𝑋)
28	1	fvexi 6856	. . . . . 6 ⊢ 𝑋 ∈ V
29	28	a1i 11	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝑋 ∈ V)
30	4, 24, 27, 29	qliftf 8754	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (Fun 𝐹 ↔ 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂))
31	5, 30	mpbid 232	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂)
32		eqid 2737	. . . . 5 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
33		fveqeq2 6851	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
34		eqeq1 2741	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → ([𝑧] ∼ = 𝑏 ↔ 𝑎 = 𝑏))
35	33, 34	imbi12d 344	. . . . . 6 ⊢ ([𝑧] ∼ = 𝑎 → (((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏) ↔ ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
36	35	ralbidv 3161	. . . . 5 ⊢ ([𝑧] ∼ = 𝑎 → (∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏) ↔ ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
37		fveq2 6842	. . . . . . . . 9 ⊢ ([𝑤] ∼ = 𝑏 → (𝐹‘[𝑤] ∼ ) = (𝐹‘𝑏))
38	37	eqeq2d 2748	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑏 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏)))
39		eqeq2 2749	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑏 → ([𝑧] ∼ = [𝑤] ∼ ↔ [𝑧] ∼ = 𝑏))
40	38, 39	imbi12d 344	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑏 → (((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ) ↔ ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏)))
41	1, 2, 3, 4	orbstaval 19253	. . . . . . . . . . . 12 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝐹‘[𝑧] ∼ ) = (𝑧 ⊕ 𝐴))
42	41	adantrr 718	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘[𝑧] ∼ ) = (𝑧 ⊕ 𝐴))
43	1, 2, 3, 4	orbstaval 19253	. . . . . . . . . . . 12 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → (𝐹‘[𝑤] ∼ ) = (𝑤 ⊕ 𝐴))
44	43	adantrl 717	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘[𝑤] ∼ ) = (𝑤 ⊕ 𝐴))
45	42, 44	eqeq12d 2753	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
46	1, 2, 3	gastacos 19251	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
47	27	adantr 480	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ∼ Er 𝑋)
48		simprl 771	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
49	47, 48	erth 8700	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ [𝑧] ∼ = [𝑤] ∼ ))
50	45, 46, 49	3bitr2d 307	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ [𝑧] ∼ = [𝑤] ∼ ))
51	50	biimpd 229	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
52	51	anassrs 467	. . . . . . 7 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
53	32, 40, 52	ectocld 8731	. . . . . 6 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ (𝑋 / ∼ )) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
54	53	ralrimiva 3130	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
55	32, 36, 54	ectocld 8731	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑎 ∈ (𝑋 / ∼ )) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
56	55	ralrimiva 3130	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀𝑎 ∈ (𝑋 / ∼ )∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
57		dff13 7210	. . 3 ⊢ (𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂 ∧ ∀𝑎 ∈ (𝑋 / ∼ )∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
58	31, 56, 57	sylanbrc 584	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂)
59		vex 3446	. . . . . . . . 9 ⊢ ℎ ∈ V
60		elecg 8690	. . . . . . . . 9 ⊢ ((ℎ ∈ V ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
61	59, 6, 60	sylancr 588	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
62	18	gaorb 19248	. . . . . . . 8 ⊢ (𝐴𝑂ℎ ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
63	61, 62	bitrdi 287	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)))
64	63	biimpa 476	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
65	64	simp3d 1145	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)
66	3	ovexi 7402	. . . . . . . . . 10 ⊢ ∼ ∈ V
67	66	ecelqsi 8718	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑋 → [𝑤] ∼ ∈ (𝑋 / ∼ ))
68	43	eqcomd 2743	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ ))
69		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑧 = [𝑤] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑤] ∼ ))
70	69	rspceeqv 3601	. . . . . . . . 9 ⊢ (([𝑤] ∼ ∈ (𝑋 / ∼ ) ∧ (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ )) → ∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧))
71	67, 68, 70	syl2an2 687	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → ∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧))
72		eqeq1 2741	. . . . . . . . 9 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → ((𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ℎ = (𝐹‘𝑧)))
73	72	rexbidv 3162	. . . . . . . 8 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → (∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
74	71, 73	syl5ibcom 245	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → ((𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
75	74	rexlimdva 3139	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
76	75	imp 406	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
77	65, 76	syldan 592	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
78	77	ralrimiva 3130	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
79		dffo3 7056	. . 3 ⊢ (𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂 ∧ ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
80	31, 78, 79	sylanbrc 584	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂)
81		df-f1o 6507	. 2 ⊢ (𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂 ∧ 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂))
82	58, 80, 81	sylanbrc 584	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ⊆ wss 3903  {cpr 4584  ⟨cop 4588   class class class wbr 5100  {copab 5162   ↦ cmpt 5181   × cxp 5630  ran crn 5633  Fun wfun 6494  ⟶wf 6496  –1-1→wf1 6497  –onto→wfo 6498  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368   Er wer 8642  [cec 8643   / cqs 8644  Basecbs 17148  SubGrpcsubg 19062   ~_QG cqg 19064   GrpAct cga 19230

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 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-ec 8647  df-qs 8651  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879  df-subg 19065  df-eqg 19067  df-ga 19231

This theorem is referenced by:  orbsta2  19255