Theorem f1omvdconj 17866

Description: Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
f1omvdconj	⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I )))

Proof of Theorem f1omvdconj

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 3737	. . . . . 6 ⊢ (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ ((𝐺 ∘ 𝐹) ∘ ◡𝐺)
2		dmss 5323	. . . . . 6 ⊢ ((((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ ((𝐺 ∘ 𝐹) ∘ ◡𝐺) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ dom ((𝐺 ∘ 𝐹) ∘ ◡𝐺))
3	1, 2	ax-mp 5	. . . . 5 ⊢ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ dom ((𝐺 ∘ 𝐹) ∘ ◡𝐺)
4		dmcoss 5385	. . . . 5 ⊢ dom ((𝐺 ∘ 𝐹) ∘ ◡𝐺) ⊆ dom ◡𝐺
5	3, 4	sstri 3612	. . . 4 ⊢ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ dom ◡𝐺
6		f1ocnv 6149	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → ◡𝐺:𝐴–1-1-onto→𝐴)
7	6	adantl 482	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ◡𝐺:𝐴–1-1-onto→𝐴)
8		f1odm 6141	. . . . 5 ⊢ (◡𝐺:𝐴–1-1-onto→𝐴 → dom ◡𝐺 = 𝐴)
9	7, 8	syl 17	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom ◡𝐺 = 𝐴)
10	5, 9	syl5sseq 3653	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ 𝐴)
11	10	sselda 3603	. 2 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I )) → 𝑥 ∈ 𝐴)
12		imassrn 5477	. . . 4 ⊢ (𝐺 “ dom (𝐹 ∖ I )) ⊆ ran 𝐺
13		f1of 6137	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺:𝐴⟶𝐴)
14	13	adantl 482	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → 𝐺:𝐴⟶𝐴)
15		frn 6053	. . . . 5 ⊢ (𝐺:𝐴⟶𝐴 → ran 𝐺 ⊆ 𝐴)
16	14, 15	syl 17	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ran 𝐺 ⊆ 𝐴)
17	12, 16	syl5ss 3614	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (𝐺 “ dom (𝐹 ∖ I )) ⊆ 𝐴)
18	17	sselda 3603	. 2 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))) → 𝑥 ∈ 𝐴)
19		simpl 473	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → 𝐹:𝐴⟶𝐴)
20		fco 6058	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐴 ∧ 𝐹:𝐴⟶𝐴) → (𝐺 ∘ 𝐹):𝐴⟶𝐴)
21	14, 19, 20	syl2anc 693	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (𝐺 ∘ 𝐹):𝐴⟶𝐴)
22		f1of 6137	. . . . . . 7 ⊢ (◡𝐺:𝐴–1-1-onto→𝐴 → ◡𝐺:𝐴⟶𝐴)
23	7, 22	syl 17	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ◡𝐺:𝐴⟶𝐴)
24		fco 6058	. . . . . 6 ⊢ (((𝐺 ∘ 𝐹):𝐴⟶𝐴 ∧ ◡𝐺:𝐴⟶𝐴) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐴⟶𝐴)
25	21, 23, 24	syl2anc 693	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐴⟶𝐴)
26		ffn 6045	. . . . 5 ⊢ (((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐴⟶𝐴 → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) Fn 𝐴)
27	25, 26	syl 17	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) Fn 𝐴)
28		fnelnfp 6443	. . . 4 ⊢ ((((𝐺 ∘ 𝐹) ∘ ◡𝐺) Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ↔ (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) ≠ 𝑥))
29	27, 28	sylan 488	. . 3 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ↔ (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) ≠ 𝑥))
30		f1ofn 6138	. . . . . . . . 9 ⊢ (◡𝐺:𝐴–1-1-onto→𝐴 → ◡𝐺 Fn 𝐴)
31	7, 30	syl 17	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ◡𝐺 Fn 𝐴)
32		fvco2 6273	. . . . . . . 8 ⊢ ((◡𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)))
33	31, 32	sylan 488	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)))
34		ffn 6045	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐴 → 𝐹 Fn 𝐴)
35	34	ad2antrr 762	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 𝐴)
36		ffvelrn 6357	. . . . . . . . 9 ⊢ ((◡𝐺:𝐴⟶𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐺‘𝑥) ∈ 𝐴)
37	23, 36	sylan 488	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (◡𝐺‘𝑥) ∈ 𝐴)
38		fvco2 6273	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐺‘𝑥) ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)) = (𝐺‘(𝐹‘(◡𝐺‘𝑥))))
39	35, 37, 38	syl2anc 693	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)) = (𝐺‘(𝐹‘(◡𝐺‘𝑥))))
40	33, 39	eqtrd 2656	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = (𝐺‘(𝐹‘(◡𝐺‘𝑥))))
41	40	eqeq1d 2624	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = 𝑥 ↔ (𝐺‘(𝐹‘(◡𝐺‘𝑥))) = 𝑥))
42		simplr 792	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐺:𝐴–1-1-onto→𝐴)
43		simpll 790	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶𝐴)
44		ffvelrn 6357	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐴 ∧ (◡𝐺‘𝑥) ∈ 𝐴) → (𝐹‘(◡𝐺‘𝑥)) ∈ 𝐴)
45	43, 37, 44	syl2anc 693	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(◡𝐺‘𝑥)) ∈ 𝐴)
46		simpr 477	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
47		f1ocnvfvb 6535	. . . . . 6 ⊢ ((𝐺:𝐴–1-1-onto→𝐴 ∧ (𝐹‘(◡𝐺‘𝑥)) ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(𝐹‘(◡𝐺‘𝑥))) = 𝑥 ↔ (◡𝐺‘𝑥) = (𝐹‘(◡𝐺‘𝑥))))
48	42, 45, 46, 47	syl3anc 1326	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(𝐹‘(◡𝐺‘𝑥))) = 𝑥 ↔ (◡𝐺‘𝑥) = (𝐹‘(◡𝐺‘𝑥))))
49	41, 48	bitrd 268	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = 𝑥 ↔ (◡𝐺‘𝑥) = (𝐹‘(◡𝐺‘𝑥))))
50	49	necon3bid 2838	. . 3 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) ≠ 𝑥 ↔ (◡𝐺‘𝑥) ≠ (𝐹‘(◡𝐺‘𝑥))))
51		necom 2847	. . . 4 ⊢ ((◡𝐺‘𝑥) ≠ (𝐹‘(◡𝐺‘𝑥)) ↔ (𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥))
52		f1of1 6136	. . . . . . 7 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺:𝐴–1-1→𝐴)
53	52	ad2antlr 763	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐺:𝐴–1-1→𝐴)
54		difss 3737	. . . . . . . . 9 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
55		dmss 5323	. . . . . . . . 9 ⊢ ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
56	54, 55	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ I ) ⊆ dom 𝐹
57		fdm 6051	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐴 → dom 𝐹 = 𝐴)
58	56, 57	syl5sseq 3653	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
59	58	ad2antrr 762	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → dom (𝐹 ∖ I ) ⊆ 𝐴)
60		f1elima 6520	. . . . . 6 ⊢ ((𝐺:𝐴–1-1→𝐴 ∧ (◡𝐺‘𝑥) ∈ 𝐴 ∧ dom (𝐹 ∖ I ) ⊆ 𝐴) → ((𝐺‘(◡𝐺‘𝑥)) ∈ (𝐺 “ dom (𝐹 ∖ I )) ↔ (◡𝐺‘𝑥) ∈ dom (𝐹 ∖ I )))
61	53, 37, 59, 60	syl3anc 1326	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(◡𝐺‘𝑥)) ∈ (𝐺 “ dom (𝐹 ∖ I )) ↔ (◡𝐺‘𝑥) ∈ dom (𝐹 ∖ I )))
62		f1ocnvfv2 6533	. . . . . . 7 ⊢ ((𝐺:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘(◡𝐺‘𝑥)) = 𝑥)
63	62	adantll 750	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐺‘(◡𝐺‘𝑥)) = 𝑥)
64	63	eleq1d 2686	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(◡𝐺‘𝑥)) ∈ (𝐺 “ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
65		fnelnfp 6443	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐺‘𝑥) ∈ 𝐴) → ((◡𝐺‘𝑥) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥)))
66	35, 37, 65	syl2anc 693	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((◡𝐺‘𝑥) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥)))
67	61, 64, 66	3bitr3rd 299	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
68	51, 67	syl5bb 272	. . 3 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((◡𝐺‘𝑥) ≠ (𝐹‘(◡𝐺‘𝑥)) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
69	29, 50, 68	3bitrd 294	. 2 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
70	11, 18, 69	eqrdav 2621	1 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I )))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∖ cdif 3571   ⊆ wss 3574   I cid 5023  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  pmtrfconj  17886  psgnunilem1  17913