Theorem orbsta 19084

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 19082	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → Fun 𝐹)
6		simpr 485	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ 𝑌)
7	6	adantr 481	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ 𝑌)
8	1	gaf 19066	. . . . . . . . . 10 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
9	8	adantr 481	. . . . . . . . 9 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
10	9	adantr 481	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
11		simpr 485	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
12	10, 11, 7	fovcdmd 7522	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ 𝑌)
13		eqid 2736	. . . . . . . 8 ⊢ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)
14		oveq1 7360	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
15	14	eqeq1d 2738	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴) ↔ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
16	15	rspcev 3579	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝑋 ∧ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
17	11, 13, 16	sylancl 586	. . . . . . 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 19078	. . . . . . 7 ⊢ (𝐴𝑂(𝑘 ⊕ 𝐴) ↔ (𝐴 ∈ 𝑌 ∧ (𝑘 ⊕ 𝐴) ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
20	7, 12, 17, 19	syl3anbrc 1343	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝐴𝑂(𝑘 ⊕ 𝐴))
21		ovex 7386	. . . . . . 7 ⊢ (𝑘 ⊕ 𝐴) ∈ V
22		elecg 8687	. . . . . . 7 ⊢ (((𝑘 ⊕ 𝐴) ∈ V ∧ 𝐴 ∈ 𝑌) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
23	21, 7, 22	sylancr 587	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
24	20, 23	mpbird 256	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂)
25	1, 2	gastacl 19080	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
26	1, 3	eqger 18971	. . . . . 6 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
27	25, 26	syl 17	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∼ Er 𝑋)
28	1	fvexi 6853	. . . . . 6 ⊢ 𝑋 ∈ V
29	28	a1i 11	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝑋 ∈ V)
30	4, 24, 27, 29	qliftf 8740	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (Fun 𝐹 ↔ 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂))
31	5, 30	mpbid 231	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂)
32		eqid 2736	. . . . 5 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
33		fveqeq2 6848	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
34		eqeq1 2740	. . . . . . 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 3172	. . . . 5 ⊢ ([𝑧] ∼ = 𝑎 → (∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏) ↔ ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
37		fveq2 6839	. . . . . . . . 9 ⊢ ([𝑤] ∼ = 𝑏 → (𝐹‘[𝑤] ∼ ) = (𝐹‘𝑏))
38	37	eqeq2d 2747	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑏 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏)))
39		eqeq2 2748	. . . . . . . 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 19083	. . . . . . . . . . . 12 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝐹‘[𝑧] ∼ ) = (𝑧 ⊕ 𝐴))
42	41	adantrr 715	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘[𝑧] ∼ ) = (𝑧 ⊕ 𝐴))
43	1, 2, 3, 4	orbstaval 19083	. . . . . . . . . . . 12 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → (𝐹‘[𝑤] ∼ ) = (𝑤 ⊕ 𝐴))
44	43	adantrl 714	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘[𝑤] ∼ ) = (𝑤 ⊕ 𝐴))
45	42, 44	eqeq12d 2752	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
46	1, 2, 3	gastacos 19081	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
47	27	adantr 481	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ∼ Er 𝑋)
48		simprl 769	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
49	47, 48	erth 8693	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ [𝑧] ∼ = [𝑤] ∼ ))
50	45, 46, 49	3bitr2d 306	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ [𝑧] ∼ = [𝑤] ∼ ))
51	50	biimpd 228	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
52	51	anassrs 468	. . . . . . 7 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
53	32, 40, 52	ectocld 8719	. . . . . 6 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ (𝑋 / ∼ )) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
54	53	ralrimiva 3141	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
55	32, 36, 54	ectocld 8719	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑎 ∈ (𝑋 / ∼ )) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
56	55	ralrimiva 3141	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀𝑎 ∈ (𝑋 / ∼ )∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
57		dff13 7198	. . 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 583	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂)
59		vex 3447	. . . . . . . . 9 ⊢ ℎ ∈ V
60		elecg 8687	. . . . . . . . 9 ⊢ ((ℎ ∈ V ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
61	59, 6, 60	sylancr 587	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
62	18	gaorb 19078	. . . . . . . 8 ⊢ (𝐴𝑂ℎ ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
63	61, 62	bitrdi 286	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)))
64	63	biimpa 477	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
65	64	simp3d 1144	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)
66	3	ovexi 7387	. . . . . . . . . 10 ⊢ ∼ ∈ V
67	66	ecelqsi 8708	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑋 → [𝑤] ∼ ∈ (𝑋 / ∼ ))
68	43	eqcomd 2742	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ ))
69		fveq2 6839	. . . . . . . . . 10 ⊢ (𝑧 = [𝑤] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑤] ∼ ))
70	69	rspceeqv 3593	. . . . . . . . 9 ⊢ (([𝑤] ∼ ∈ (𝑋 / ∼ ) ∧ (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ )) → ∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧))
71	67, 68, 70	syl2an2 684	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → ∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧))
72		eqeq1 2740	. . . . . . . . 9 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → ((𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ℎ = (𝐹‘𝑧)))
73	72	rexbidv 3173	. . . . . . . 8 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → (∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
74	71, 73	syl5ibcom 244	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → ((𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
75	74	rexlimdva 3150	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
76	75	imp 407	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
77	65, 76	syldan 591	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
78	77	ralrimiva 3141	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
79		dffo3 7048	. . 3 ⊢ (𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂 ∧ ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
80	31, 78, 79	sylanbrc 583	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂)
81		df-f1o 6500	. 2 ⊢ (𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂 ∧ 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂))
82	58, 80, 81	sylanbrc 583	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3062  ∃wrex 3071  {crab 3405  Vcvv 3443   ⊆ wss 3908  {cpr 4586  ⟨cop 4590   class class class wbr 5103  {copab 5165   ↦ cmpt 5186   × cxp 5629  ran crn 5632  Fun wfun 6487  ⟶wf 6489  –1-1→wf1 6490  –onto→wfo 6491  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7353   Er wer 8641  [cec 8642   / cqs 8643  Basecbs 17075  SubGrpcsubg 18913   ~_QG cqg 18915   GrpAct cga 19060

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-er 8644  df-ec 8646  df-qs 8650  df-map 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-2 12212  df-sets 17028  df-slot 17046  df-ndx 17058  df-base 17076  df-ress 17105  df-plusg 17138  df-0g 17315  df-mgm 18489  df-sgrp 18538  df-mnd 18549  df-grp 18743  df-minusg 18744  df-subg 18916  df-eqg 18918  df-ga 19061

This theorem is referenced by:  orbsta2  19085