Theorem orbsta 17569

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 17567	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → Fun 𝐹)
6		simpr 476	. . . . . . . 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 17551	. . . . . . . . . 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 476	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
12	10, 11, 7	fovrnd 6704	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ 𝑌)
13		eqid 2610	. . . . . . . 8 ⊢ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)
14		oveq1 6556	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
15	14	eqeq1d 2612	. . . . . . . . 9 ⊢ (ℎ = 𝑘 → ((ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴) ↔ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
16	15	rspcev 3282	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝑋 ∧ (𝑘 ⊕ 𝐴) = (𝑘 ⊕ 𝐴)) → ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
17	11, 13, 16	sylancl 693	. . . . . . 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 17563	. . . . . . 7 ⊢ (𝐴𝑂(𝑘 ⊕ 𝐴) ↔ (𝐴 ∈ 𝑌 ∧ (𝑘 ⊕ 𝐴) ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴)))
20	7, 12, 17, 19	syl3anbrc 1239	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → 𝐴𝑂(𝑘 ⊕ 𝐴))
21		ovex 6577	. . . . . . 7 ⊢ (𝑘 ⊕ 𝐴) ∈ V
22		elecg 7672	. . . . . . 7 ⊢ (((𝑘 ⊕ 𝐴) ∈ V ∧ 𝐴 ∈ 𝑌) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
23	21, 7, 22	sylancr 694	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂 ↔ 𝐴𝑂(𝑘 ⊕ 𝐴)))
24	20, 23	mpbird 246	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) ∈ [𝐴]𝑂)
25	1, 2	gastacl 17565	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
26	1, 3	eqger 17467	. . . . . 6 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
27	25, 26	syl 17	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∼ Er 𝑋)
28		fvex 6113	. . . . . . 7 ⊢ (Base‘𝐺) ∈ V
29	1, 28	eqeltri 2684	. . . . . 6 ⊢ 𝑋 ∈ V
30	29	a1i 11	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝑋 ∈ V)
31	4, 24, 27, 30	qliftf 7722	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (Fun 𝐹 ↔ 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂))
32	5, 31	mpbid 221	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂)
33		eqid 2610	. . . . 5 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
34		fveq2 6103	. . . . . . . 8 ⊢ ([𝑧] ∼ = 𝑎 → (𝐹‘[𝑧] ∼ ) = (𝐹‘𝑎))
35	34	eqeq1d 2612	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
36		eqeq1 2614	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → ([𝑧] ∼ = 𝑏 ↔ 𝑎 = 𝑏))
37	35, 36	imbi12d 333	. . . . . 6 ⊢ ([𝑧] ∼ = 𝑎 → (((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏) ↔ ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
38	37	ralbidv 2969	. . . . 5 ⊢ ([𝑧] ∼ = 𝑎 → (∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏) ↔ ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
39		fveq2 6103	. . . . . . . . 9 ⊢ ([𝑤] ∼ = 𝑏 → (𝐹‘[𝑤] ∼ ) = (𝐹‘𝑏))
40	39	eqeq2d 2620	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑏 → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏)))
41		eqeq2 2621	. . . . . . . 8 ⊢ ([𝑤] ∼ = 𝑏 → ([𝑧] ∼ = [𝑤] ∼ ↔ [𝑧] ∼ = 𝑏))
42	40, 41	imbi12d 333	. . . . . . 7 ⊢ ([𝑤] ∼ = 𝑏 → (((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ) ↔ ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏)))
43	1, 2, 3, 4	orbstaval 17568	. . . . . . . . . . . 12 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → (𝐹‘[𝑧] ∼ ) = (𝑧 ⊕ 𝐴))
44	43	adantrr 749	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘[𝑧] ∼ ) = (𝑧 ⊕ 𝐴))
45	1, 2, 3, 4	orbstaval 17568	. . . . . . . . . . . 12 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → (𝐹‘[𝑤] ∼ ) = (𝑤 ⊕ 𝐴))
46	45	adantrl 748	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝐹‘[𝑤] ∼ ) = (𝑤 ⊕ 𝐴))
47	44, 46	eqeq12d 2625	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
48	1, 2, 3	gastacos 17566	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ (𝑧 ⊕ 𝐴) = (𝑤 ⊕ 𝐴)))
49	27	adantr 480	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ∼ Er 𝑋)
50		simprl 790	. . . . . . . . . . 11 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
51	49, 50	erth 7678	. . . . . . . . . 10 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ∼ 𝑤 ↔ [𝑧] ∼ = [𝑤] ∼ ))
52	47, 48, 51	3bitr2d 295	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) ↔ [𝑧] ∼ = [𝑤] ∼ ))
53	52	biimpd 218	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
54	53	anassrs 678	. . . . . . 7 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘[𝑤] ∼ ) → [𝑧] ∼ = [𝑤] ∼ ))
55	33, 42, 54	ectocld 7701	. . . . . 6 ⊢ (((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ (𝑋 / ∼ )) → ((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
56	55	ralrimiva 2949	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑧 ∈ 𝑋) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘[𝑧] ∼ ) = (𝐹‘𝑏) → [𝑧] ∼ = 𝑏))
57	33, 38, 56	ectocld 7701	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑎 ∈ (𝑋 / ∼ )) → ∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
58	57	ralrimiva 2949	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀𝑎 ∈ (𝑋 / ∼ )∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
59		dff13 6416	. . 3 ⊢ (𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂 ∧ ∀𝑎 ∈ (𝑋 / ∼ )∀𝑏 ∈ (𝑋 / ∼ )((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
60	32, 58, 59	sylanbrc 695	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂)
61		vex 3176	. . . . . . . . 9 ⊢ ℎ ∈ V
62		elecg 7672	. . . . . . . . 9 ⊢ ((ℎ ∈ V ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
63	61, 6, 62	sylancr 694	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ 𝐴𝑂ℎ))
64	18	gaorb 17563	. . . . . . . 8 ⊢ (𝐴𝑂ℎ ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
65	63, 64	syl6bb 275	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (ℎ ∈ [𝐴]𝑂 ↔ (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)))
66	65	biimpa 500	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → (𝐴 ∈ 𝑌 ∧ ℎ ∈ 𝑌 ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ))
67	66	simp3d 1068	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ)
68		ovex 6577	. . . . . . . . . . . 12 ⊢ (𝐺 ~QG 𝐻) ∈ V
69	3, 68	eqeltri 2684	. . . . . . . . . . 11 ⊢ ∼ ∈ V
70	69	ecelqsi 7690	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑋 → [𝑤] ∼ ∈ (𝑋 / ∼ ))
71	70	adantl 481	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → [𝑤] ∼ ∈ (𝑋 / ∼ ))
72	45	eqcomd 2616	. . . . . . . . 9 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ ))
73		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑧 = [𝑤] ∼ → (𝐹‘𝑧) = (𝐹‘[𝑤] ∼ ))
74	73	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑧 = [𝑤] ∼ → ((𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ )))
75	74	rspcev 3282	. . . . . . . . 9 ⊢ (([𝑤] ∼ ∈ (𝑋 / ∼ ) ∧ (𝑤 ⊕ 𝐴) = (𝐹‘[𝑤] ∼ )) → ∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧))
76	71, 72, 75	syl2anc 691	. . . . . . . 8 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → ∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧))
77		eqeq1 2614	. . . . . . . . 9 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → ((𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ℎ = (𝐹‘𝑧)))
78	77	rexbidv 3034	. . . . . . . 8 ⊢ ((𝑤 ⊕ 𝐴) = ℎ → (∃𝑧 ∈ (𝑋 / ∼ )(𝑤 ⊕ 𝐴) = (𝐹‘𝑧) ↔ ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
79	76, 78	syl5ibcom 234	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑤 ∈ 𝑋) → ((𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
80	79	rexlimdva 3013	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → (∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
81	80	imp 444	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ∃𝑤 ∈ 𝑋 (𝑤 ⊕ 𝐴) = ℎ) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
82	67, 81	syldan 486	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ ℎ ∈ [𝐴]𝑂) → ∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
83	82	ralrimiva 2949	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧))
84		dffo3 6282	. . 3 ⊢ (𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )⟶[𝐴]𝑂 ∧ ∀ℎ ∈ [ 𝐴]𝑂∃𝑧 ∈ (𝑋 / ∼ )ℎ = (𝐹‘𝑧)))
85	32, 83, 84	sylanbrc 695	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂)
86		df-f1o 5811	. 2 ⊢ (𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂 ↔ (𝐹:(𝑋 / ∼ )–1-1→[𝐴]𝑂 ∧ 𝐹:(𝑋 / ∼ )–onto→[𝐴]𝑂))
87	60, 85, 86	sylanbrc 695	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  {cpr 4127  ⟨cop 4131   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  ran crn 5039  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   Er wer 7626  [cec 7627   / cqs 7628  Basecbs 15695  SubGrpcsubg 17411   ~_QG cqg 17413   GrpAct cga 17545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-eqg 17416  df-ga 17546

This theorem is referenced by:  orbsta2  17570