Theorem f1omvdconj 19669

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 4071	. . . . . 6 ⊢ (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ ((𝐺 ∘ 𝐹) ∘ ◡𝐺)
2		dmss 5885	. . . . . 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 5958	. . . . 5 ⊢ dom ((𝐺 ∘ 𝐹) ∘ ◡𝐺) ⊆ dom ◡𝐺
5	3, 4	sstri 3918	. . . 4 ⊢ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ dom ◡𝐺
6		f1ocnv 6842	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → ◡𝐺:𝐴–1-1-onto→𝐴)
7	6	adantl 487	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ◡𝐺:𝐴–1-1-onto→𝐴)
8		f1odm 6833	. . . . 5 ⊢ (◡𝐺:𝐴–1-1-onto→𝐴 → dom ◡𝐺 = 𝐴)
9	7, 8	syl 18	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom ◡𝐺 = 𝐴)
10	5, 9	sseqtrid 3951	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ 𝐴)
11	10	sselda 3909	. 2 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I )) → 𝑥 ∈ 𝐴)
12		imassrn 6200	. . . 4 ⊢ (𝐺 “ dom (𝐹 ∖ I )) ⊆ ran 𝐺
13		f1of 6829	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺:𝐴⟶𝐴)
14	13	adantl 487	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → 𝐺:𝐴⟶𝐴)
15	14	frnd 6723	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ran 𝐺 ⊆ 𝐴)
16	12, 15	sstrid 3920	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (𝐺 “ dom (𝐹 ∖ I )) ⊆ 𝐴)
17	16	sselda 3909	. 2 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))) → 𝑥 ∈ 𝐴)
18		simpl 488	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → 𝐹:𝐴⟶𝐴)
19		fco 6739	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐴 ∧ 𝐹:𝐴⟶𝐴) → (𝐺 ∘ 𝐹):𝐴⟶𝐴)
20	14, 18, 19	syl2anc 596	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (𝐺 ∘ 𝐹):𝐴⟶𝐴)
21		f1of 6829	. . . . . . 7 ⊢ (◡𝐺:𝐴–1-1-onto→𝐴 → ◡𝐺:𝐴⟶𝐴)
22	7, 21	syl 18	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ◡𝐺:𝐴⟶𝐴)
23		fco 6739	. . . . . 6 ⊢ (((𝐺 ∘ 𝐹):𝐴⟶𝐴 ∧ ◡𝐺:𝐴⟶𝐴) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐴⟶𝐴)
24	20, 22, 23	syl2anc 596	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐴⟶𝐴)
25	24	ffnd 6715	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) Fn 𝐴)
26		fnelnfp 7186	. . . 4 ⊢ ((((𝐺 ∘ 𝐹) ∘ ◡𝐺) Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ↔ (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) ≠ 𝑥))
27	25, 26	sylan 592	. . 3 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ↔ (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) ≠ 𝑥))
28		f1ofn 6830	. . . . . . . . 9 ⊢ (◡𝐺:𝐴–1-1-onto→𝐴 → ◡𝐺 Fn 𝐴)
29	7, 28	syl 18	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ◡𝐺 Fn 𝐴)
30		fvco2 6987	. . . . . . . 8 ⊢ ((◡𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)))
31	29, 30	sylan 592	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)))
32		ffn 6714	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐴 → 𝐹 Fn 𝐴)
33	32	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 𝐴)
34		ffvelcdm 7086	. . . . . . . . 9 ⊢ ((◡𝐺:𝐴⟶𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐺‘𝑥) ∈ 𝐴)
35	22, 34	sylan 592	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (◡𝐺‘𝑥) ∈ 𝐴)
36		fvco2 6987	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐺‘𝑥) ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)) = (𝐺‘(𝐹‘(◡𝐺‘𝑥))))
37	33, 35, 36	syl2anc 596	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)) = (𝐺‘(𝐹‘(◡𝐺‘𝑥))))
38	31, 37	eqtrd 2795	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = (𝐺‘(𝐹‘(◡𝐺‘𝑥))))
39	38	eqeq1d 2762	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = 𝑥 ↔ (𝐺‘(𝐹‘(◡𝐺‘𝑥))) = 𝑥))
40		simplr 781	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐺:𝐴–1-1-onto→𝐴)
41		simpll 779	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶𝐴)
42		ffvelcdm 7086	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐴 ∧ (◡𝐺‘𝑥) ∈ 𝐴) → (𝐹‘(◡𝐺‘𝑥)) ∈ 𝐴)
43	41, 35, 42	syl2anc 596	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(◡𝐺‘𝑥)) ∈ 𝐴)
44		simpr 490	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
45		f1ocnvfvb 7291	. . . . . 6 ⊢ ((𝐺:𝐴–1-1-onto→𝐴 ∧ (𝐹‘(◡𝐺‘𝑥)) ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(𝐹‘(◡𝐺‘𝑥))) = 𝑥 ↔ (◡𝐺‘𝑥) = (𝐹‘(◡𝐺‘𝑥))))
46	40, 43, 44, 45	syl3anc 1398	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(𝐹‘(◡𝐺‘𝑥))) = 𝑥 ↔ (◡𝐺‘𝑥) = (𝐹‘(◡𝐺‘𝑥))))
47	39, 46	bitrd 282	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = 𝑥 ↔ (◡𝐺‘𝑥) = (𝐹‘(◡𝐺‘𝑥))))
48	47	necon3bid 2999	. . 3 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) ≠ 𝑥 ↔ (◡𝐺‘𝑥) ≠ (𝐹‘(◡𝐺‘𝑥))))
49		necom 3008	. . . 4 ⊢ ((◡𝐺‘𝑥) ≠ (𝐹‘(◡𝐺‘𝑥)) ↔ (𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥))
50		f1of1 6828	. . . . . . 7 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺:𝐴–1-1→𝐴)
51	50	ad2antlr 740	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐺:𝐴–1-1→𝐴)
52		difss 4071	. . . . . . . . 9 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
53		dmss 5885	. . . . . . . . 9 ⊢ ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
54	52, 53	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ I ) ⊆ dom 𝐹
55		fdm 6724	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐴 → dom 𝐹 = 𝐴)
56	54, 55	sseqtrid 3951	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
57	56	ad2antrr 739	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → dom (𝐹 ∖ I ) ⊆ 𝐴)
58		f1elima 7271	. . . . . 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 )))
59	51, 35, 57, 58	syl3anc 1398	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(◡𝐺‘𝑥)) ∈ (𝐺 “ dom (𝐹 ∖ I )) ↔ (◡𝐺‘𝑥) ∈ dom (𝐹 ∖ I )))
60		f1ocnvfv2 7289	. . . . . . 7 ⊢ ((𝐺:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘(◡𝐺‘𝑥)) = 𝑥)
61	60	adantll 727	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐺‘(◡𝐺‘𝑥)) = 𝑥)
62	61	eleq1d 2845	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(◡𝐺‘𝑥)) ∈ (𝐺 “ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
63		fnelnfp 7186	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (◡𝐺‘𝑥) ∈ 𝐴) → ((◡𝐺‘𝑥) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥)))
64	33, 35, 63	syl2anc 596	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((◡𝐺‘𝑥) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥)))
65	59, 62, 64	3bitr3rd 313	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
66	49, 65	bitrid 286	. . 3 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → ((◡𝐺‘𝑥) ≠ (𝐹‘(◡𝐺‘𝑥)) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
67	27, 48, 66	3bitrd 308	. 2 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
68	11, 17, 67	eqrdav 2759	1 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3896   ∖ cdif 4044   I cid 5545  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654   ∘ ccom 5655   Fn wfn 6538  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551

This theorem is used by:  pmtrfconj  19689  psgnunilem1  19716