Theorem orbsta 19387

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 19385	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → Fun 𝐹)
6		simpr 489	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ 𝑌)
7	6	adantr 485	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ 𝑌)
8	1	gaf 19369	. . . . . . . . . 10 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
9	8	adantr 485	. . . . . . . . 9 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
10	9	adantr 485	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
11		simpr 489	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
12	10, 11, 7	fovcdmd 7582	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ 𝑌)
13		eqid 2763	. . . . . . . 8 ⊢ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)
14		oveq1 7417	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
15	14	eqeq1d 2765	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴) ↔ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
16	15	rspcev 3581	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝑋 ∧ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
17	11, 13, 16	sylancl 597	. . . . . . 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 19381	. . . . . . 7 ⊢ (𝐴𝑂(𝑘 ⊕ 𝐴) ↔ (𝐴 ∈ 𝑌 ∧ (𝑘 ⊕ 𝐴) ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
20	7, 12, 17, 19	syl3anbrc 1362	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝐴𝑂(𝑘 ⊕ 𝐴))
21		ovex 7443	. . . . . . 7 ⊢ (𝑘 ⊕ 𝐴) ∈ V
22		elecg 8735	. . . . . . 7 ⊢ (((𝑘 ⊕ 𝐴) ∈ V ∧ 𝐴 ∈ 𝑌) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
23	21, 7, 22	sylancr 598	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
24	20, 23	mpbird 260	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂)
25	1, 2	gastacl 19383	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
26	1, 3	eqger 19250	. . . . . 6 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
27	25, 26	syl 18	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∼ Er 𝑋)
28	1	fvexi 6895	. . . . . 6 ⊢ 𝑋 ∈ V
29	28	a1i 11	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝑋 ∈ V)
30	4, 24, 27, 29	qliftf 8799	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (Fun 𝐹 ↔ 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂))
31	5, 30	mpbid 235	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂)
32		eqid 2763	. . . . 5 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
33		fveqeq2 6890	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
34		eqeq1 2767	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → ([𝑧] ∼ = 𝑏 ↔ 𝑎 = 𝑏))
35	33, 34	imbi12d 347	. . . . . 6 ⊢ ([𝑧] ∼ = 𝑎 → (((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏) ↔ ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
36	35	ralbidv 3188	. . . . 5 ⊢ ([𝑧] ∼ = 𝑎 → (∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏) ↔ ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
37		fveq2 6881	. . . . . . . . 9 ⊢ ([𝑤] ∼ = 𝑏 → (𝐹‘[𝑤] ∼ ) = (𝐹‘𝑏))
38	37	eqeq2d 2774	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑏 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏)))
39		eqeq2 2775	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑏 → ([𝑧] ∼ = [𝑤] ∼ ↔ [𝑧] ∼ = 𝑏))
40	38, 39	imbi12d 347	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑏 → (((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ) ↔ ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏)))
41	1, 2, 3, 4	orbstaval 19386	. . . . . . . . . . . 12 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝐹‘[𝑧] ∼ ) = (𝑧 ⊕ 𝐴))
42	41	adantrr 729	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘[𝑧] ∼ ) = (𝑧 ⊕ 𝐴))
43	1, 2, 3, 4	orbstaval 19386	. . . . . . . . . . . 12 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → (𝐹‘[𝑤] ∼ ) = (𝑤 ⊕ 𝐴))
44	43	adantrl 728	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘[𝑤] ∼ ) = (𝑤 ⊕ 𝐴))
45	42, 44	eqeq12d 2779	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
46	1, 2, 3	gastacos 19384	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
47	27	adantr 485	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ∼ Er 𝑋)
48		simprl 782	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
49	47, 48	erth 8745	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ [𝑧] ∼ = [𝑤] ∼ ))
50	45, 46, 49	3bitr2d 310	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ [𝑧] ∼ = [𝑤] ∼ ))
51	50	biimpd 232	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
52	51	anassrs 472	. . . . . . 7 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
53	32, 40, 52	ectocld 8776	. . . . . 6 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ (𝑋 / ∼ )) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
54	53	ralrimiva 3157	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
55	32, 36, 54	ectocld 8776	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑎 ∈ (𝑋 / ∼ )) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
56	55	ralrimiva 3157	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀𝑎 ∈ (𝑋 / ∼ )∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
57		dff13 7252	. . 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 594	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂)
59		vex 3459	. . . . . . . . 9 ⊢ ℎ ∈ V
60		elecg 8735	. . . . . . . . 9 ⊢ ((ℎ ∈ V ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
61	59, 6, 60	sylancr 598	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
62	18	gaorb 19381	. . . . . . . 8 ⊢ (𝐴𝑂ℎ ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
63	61, 62	bitrdi 290	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)))
64	63	biimpa 481	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
65	64	simp3d 1162	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)
66	3	ovexi 7444	. . . . . . . . . 10 ⊢ ∼ ∈ V
67	66	ecelqsi 8763	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑋 → [𝑤] ∼ ∈ (𝑋 / ∼ ))
68	43	eqcomd 2769	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ ))
69		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑧 = [𝑤] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑤] ∼ ))
70	69	rspceeqv 3604	. . . . . . . . 9 ⊢ (([𝑤] ∼ ∈ (𝑋 / ∼ ) ∧ (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ )) → ∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧))
71	67, 68, 70	syl2an2 698	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → ∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧))
72		eqeq1 2767	. . . . . . . . 9 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → ((𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ℎ = (𝐹‘𝑧)))
73	72	rexbidv 3189	. . . . . . . 8 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → (∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
74	71, 73	syl5ibcom 248	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → ((𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
75	74	rexlimdva 3166	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
76	75	imp 411	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
77	65, 76	syldan 602	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
78	77	ralrimiva 3157	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
79		dffo3 7097	. . 3 ⊢ (𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂 ∧ ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
80	31, 78, 79	sylanbrc 594	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂)
81		df-f1o 6543	. 2 ⊢ (𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂 ∧ 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂))
82	58, 80, 81	sylanbrc 594	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3905  {cpr 4591  ⟨cop 4595   class class class wbr 5109  {copab 5173   ↦ cmpt 5192   × cxp 5659  ran crn 5662  Fun wfun 6530  ⟶wf 6532  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   Er wer 8687  [cec 8688   / cqs 8689  Basecbs 17273  SubGrpcsubg 19190   ~_QG cqg 19192   GrpAct cga 19363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-ec 8692  df-qs 8696  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-2 12307  df-sets 17228  df-slot 17246  df-ndx 17258  df-base 17274  df-ress 17295  df-plusg 17327  df-0g 17498  df-mgm 18702  df-sgrp 18781  df-mnd 18797  df-grp 19007  df-minusg 19008  df-subg 19193  df-eqg 19195  df-ga 19364

This theorem is used by:  orbsta2  19388