Theorem orbsta 18445

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 18443	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → Fun 𝐹)
6		simpr 487	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐴 ∈ 𝑌)
7	6	adantr 483	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ 𝑌)
8	1	gaf 18427	. . . . . . . . . 10 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
9	8	adantr 483	. . . . . . . . 9 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
10	9	adantr 483	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
11		simpr 487	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
12	10, 11, 7	fovrnd 7322	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ 𝑌)
13		eqid 2823	. . . . . . . 8 ⊢ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)
14		oveq1 7165	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
15	14	eqeq1d 2825	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴) ↔ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
16	15	rspcev 3625	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝑋 ∧ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
17	11, 13, 16	sylancl 588	. . . . . . 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 18439	. . . . . . 7 ⊢ (𝐴𝑂(𝑘 ⊕ 𝐴) ↔ (𝐴 ∈ 𝑌 ∧ (𝑘 ⊕ 𝐴) ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
20	7, 12, 17, 19	syl3anbrc 1339	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝐴𝑂(𝑘 ⊕ 𝐴))
21		ovex 7191	. . . . . . 7 ⊢ (𝑘 ⊕ 𝐴) ∈ V
22		elecg 8334	. . . . . . 7 ⊢ (((𝑘 ⊕ 𝐴) ∈ V ∧ 𝐴 ∈ 𝑌) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
23	21, 7, 22	sylancr 589	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
24	20, 23	mpbird 259	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂)
25	1, 2	gastacl 18441	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
26	1, 3	eqger 18332	. . . . . 6 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
27	25, 26	syl 17	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∼ Er 𝑋)
28	1	fvexi 6686	. . . . . 6 ⊢ 𝑋 ∈ V
29	28	a1i 11	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝑋 ∈ V)
30	4, 24, 27, 29	qliftf 8387	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (Fun 𝐹 ↔ 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂))
31	5, 30	mpbid 234	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂)
32		eqid 2823	. . . . 5 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
33		fveqeq2 6681	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
34		eqeq1 2827	. . . . . . 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 3199	. . . . 5 ⊢ ([𝑧] ∼ = 𝑎 → (∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏) ↔ ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
37		fveq2 6672	. . . . . . . . 9 ⊢ ([𝑤] ∼ = 𝑏 → (𝐹‘[𝑤] ∼ ) = (𝐹‘𝑏))
38	37	eqeq2d 2834	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑏 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏)))
39		eqeq2 2835	. . . . . . . 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 18444	. . . . . . . . . . . 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 18444	. . . . . . . . . . . 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 2839	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
46	1, 2, 3	gastacos 18442	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
47	27	adantr 483	. . . . . . . . . . 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 8340	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ [𝑧] ∼ = [𝑤] ∼ ))
50	45, 46, 49	3bitr2d 309	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ [𝑧] ∼ = [𝑤] ∼ ))
51	50	biimpd 231	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
52	51	anassrs 470	. . . . . . 7 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
53	32, 40, 52	ectocld 8366	. . . . . 6 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ (𝑋 / ∼ )) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
54	53	ralrimiva 3184	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
55	32, 36, 54	ectocld 8366	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑎 ∈ (𝑋 / ∼ )) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
56	55	ralrimiva 3184	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀𝑎 ∈ (𝑋 / ∼ )∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
57		dff13 7015	. . 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 585	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂)
59		vex 3499	. . . . . . . . 9 ⊢ ℎ ∈ V
60		elecg 8334	. . . . . . . . 9 ⊢ ((ℎ ∈ V ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
61	59, 6, 60	sylancr 589	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
62	18	gaorb 18439	. . . . . . . 8 ⊢ (𝐴𝑂ℎ ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
63	61, 62	syl6bb 289	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)))
64	63	biimpa 479	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
65	64	simp3d 1140	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)
66	3	ovexi 7192	. . . . . . . . . 10 ⊢ ∼ ∈ V
67	66	ecelqsi 8355	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑋 → [𝑤] ∼ ∈ (𝑋 / ∼ ))
68	43	eqcomd 2829	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ ))
69		fveq2 6672	. . . . . . . . . 10 ⊢ (𝑧 = [𝑤] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑤] ∼ ))
70	69	rspceeqv 3640	. . . . . . . . 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 2827	. . . . . . . . 9 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → ((𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ℎ = (𝐹‘𝑧)))
73	72	rexbidv 3299	. . . . . . . 8 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → (∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
74	71, 73	syl5ibcom 247	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → ((𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
75	74	rexlimdva 3286	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
76	75	imp 409	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
77	65, 76	syldan 593	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
78	77	ralrimiva 3184	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
79		dffo3 6870	. . 3 ⊢ (𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂 ∧ ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
80	31, 78, 79	sylanbrc 585	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂)
81		df-f1o 6364	. 2 ⊢ (𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂 ∧ 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂))
82	58, 80, 81	sylanbrc 585	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  {crab 3144  Vcvv 3496   ⊆ wss 3938  {cpr 4571  ⟨cop 4575   class class class wbr 5068  {copab 5130   ↦ cmpt 5148   × cxp 5555  ran crn 5558  Fun wfun 6351  ⟶wf 6353  –1-1→wf1 6354  –onto→wfo 6355  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158   Er wer 8288  [cec 8289   / cqs 8290  Basecbs 16485  SubGrpcsubg 18275   ~_QG cqg 18277   GrpAct cga 18421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-ec 8293  df-qs 8297  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109  df-subg 18278  df-eqg 18280  df-ga 18422

This theorem is referenced by:  orbsta2  18446