Theorem f1omvdconj 19711

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⟶1-1onto𝐴) → 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 4075	. . . . . 6 ⊢ (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ ((𝐺 ∘ 𝐹) ∘ ◡𝐺)
2		dmss 5896	. . . . . 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 5969	. . . . 5 ⊢ dom ((𝐺 ∘ 𝐹) ∘ ◡𝐺) ⊆ dom ◡𝐺
5	3, 4	sstri 3922	. . . 4 ⊢ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ dom ◡𝐺
6		f1ocnv 6867	. . . . . 6 ⊢ (𝐺:𝐴⟶1-1onto𝐴 → ◡𝐺:𝐴⟶1-1onto𝐴)
7	6	adantl 487	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ◡𝐺:𝐴⟶1-1onto𝐴)
8		f1odm 6858	. . . . 5 ⊢ (◡𝐺:𝐴⟶1-1onto𝐴 → dom ◡𝐺 = 𝐴)
9	7, 8	syl 18	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → dom ◡𝐺 = 𝐴)
10	5, 9	sseqtrid 3955	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ⊆ 𝐴)
11	10	sselda 3913	. 2 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I )) → 𝑥 ∈ 𝐴)
12		imassrn 6216	. . . 4 ⊢ (𝐺 “ dom (𝐹 ∖ I )) ⊆ ran 𝐺
13		f1of 6854	. . . . . 6 ⊢ (𝐺:𝐴⟶1-1onto𝐴 → 𝐺:𝐴⟶𝐴)
14	13	adantl 487	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → 𝐺:𝐴⟶𝐴)
15	14	frnd 6748	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ran 𝐺 ⊆ 𝐴)
16	12, 15	sstrid 3924	. . 3 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → (𝐺 “ dom (𝐹 ∖ I )) ⊆ 𝐴)
17	16	sselda 3913	. 2 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))) → 𝑥 ∈ 𝐴)
18		simpl 488	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → 𝐹:𝐴⟶𝐴)
19		fco 6764	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐴 ∧ 𝐹:𝐴⟶𝐴) → (𝐺 ∘ 𝐹):𝐴⟶𝐴)
20	14, 18, 19	syl2anc 596	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → (𝐺 ∘ 𝐹):𝐴⟶𝐴)
21		f1of 6854	. . . . . . 7 ⊢ (◡𝐺:𝐴⟶1-1onto𝐴 → ◡𝐺:𝐴⟶𝐴)
22	7, 21	syl 18	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ◡𝐺:𝐴⟶𝐴)
23		fco 6764	. . . . . 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⟶1-1onto𝐴) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺):𝐴⟶𝐴)
25	24	ffnd 6740	. . . 4 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ((𝐺 ∘ 𝐹) ∘ ◡𝐺) Fn 𝐴)
26		fnelnfp 7213	. . . 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⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ↔ (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) ≠ 𝑥))
28		f1ofn 6855	. . . . . . . . 9 ⊢ (◡𝐺:𝐴⟶1-1onto𝐴 → ◡𝐺 Fn 𝐴)
29	7, 28	syl 18	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ◡𝐺 Fn 𝐴)
30		fvco2 7013	. . . . . . . 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⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)))
32		ffn 6739	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐴 → 𝐹 Fn 𝐴)
33	32	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 𝐴)
34		ffvelcdm 7113	. . . . . . . . 9 ⊢ ((◡𝐺:𝐴⟶𝐴 ∧ 𝑥 ∈ 𝐴) → (◡𝐺‘𝑥) ∈ 𝐴)
35	22, 34	sylan 592	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → (◡𝐺‘𝑥) ∈ 𝐴)
36		fvco2 7013	. . . . . . . 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⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘(◡𝐺‘𝑥)) = (𝐺‘(𝐹‘(◡𝐺‘𝑥))))
38	31, 37	eqtrd 2798	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → (((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = (𝐺‘(𝐹‘(◡𝐺‘𝑥))))
39	38	eqeq1d 2765	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = 𝑥 ↔ (𝐺‘(𝐹‘(◡𝐺‘𝑥))) = 𝑥))
40		simplr 781	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐺:𝐴⟶1-1onto𝐴)
41		simpll 779	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐴⟶𝐴)
42		ffvelcdm 7113	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐴 ∧ (◡𝐺‘𝑥) ∈ 𝐴) → (𝐹‘(◡𝐺‘𝑥)) ∈ 𝐴)
43	41, 35, 42	syl2anc 596	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(◡𝐺‘𝑥)) ∈ 𝐴)
44		simpr 490	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
45		f1ocnvfvb 7318	. . . . . 6 ⊢ ((𝐺:𝐴⟶1-1onto𝐴 ∧ (𝐹‘(◡𝐺‘𝑥)) ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(𝐹‘(◡𝐺‘𝑥))) = 𝑥 ↔ (◡𝐺‘𝑥) = (𝐹‘(◡𝐺‘𝑥))))
46	40, 43, 44, 45	syl3anc 1399	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(𝐹‘(◡𝐺‘𝑥))) = 𝑥 ↔ (◡𝐺‘𝑥) = (𝐹‘(◡𝐺‘𝑥))))
47	39, 46	bitrd 282	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) = 𝑥 ↔ (◡𝐺‘𝑥) = (𝐹‘(◡𝐺‘𝑥))))
48	47	necon3bid 3003	. . 3 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((((𝐺 ∘ 𝐹) ∘ ◡𝐺)‘𝑥) ≠ 𝑥 ↔ (◡𝐺‘𝑥) ≠ (𝐹‘(◡𝐺‘𝑥))))
49		necom 3012	. . . 4 ⊢ ((◡𝐺‘𝑥) ≠ (𝐹‘(◡𝐺‘𝑥)) ↔ (𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥))
50		f1of1 6853	. . . . . . 7 ⊢ (𝐺:𝐴⟶1-1onto𝐴 → 𝐺:𝐴⟶1-1𝐴)
51	50	ad2antlr 740	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐺:𝐴⟶1-1𝐴)
52		difss 4075	. . . . . . . . 9 ⊢ (𝐹 ∖ I ) ⊆ 𝐹
53		dmss 5896	. . . . . . . . 9 ⊢ ((𝐹 ∖ I ) ⊆ 𝐹 → dom (𝐹 ∖ I ) ⊆ dom 𝐹)
54	52, 53	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ I ) ⊆ dom 𝐹
55		fdm 6749	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐴 → dom 𝐹 = 𝐴)
56	54, 55	sseqtrid 3955	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐴 → dom (𝐹 ∖ I ) ⊆ 𝐴)
57	56	ad2antrr 739	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → dom (𝐹 ∖ I ) ⊆ 𝐴)
58		f1elima 7298	. . . . . 6 ⊢ ((𝐺:𝐴⟶1-1𝐴 ∧ (◡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 1399	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(◡𝐺‘𝑥)) ∈ (𝐺 “ dom (𝐹 ∖ I )) ↔ (◡𝐺‘𝑥) ∈ dom (𝐹 ∖ I )))
60		f1ocnvfv2 7316	. . . . . . 7 ⊢ ((𝐺:𝐴⟶1-1onto𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘(◡𝐺‘𝑥)) = 𝑥)
61	60	adantll 727	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐺‘(◡𝐺‘𝑥)) = 𝑥)
62	61	eleq1d 2849	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘(◡𝐺‘𝑥)) ∈ (𝐺 “ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
63		fnelnfp 7213	. . . . . 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⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((◡𝐺‘𝑥) ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥)))
65	59, 62, 64	3bitr3rd 313	. . . 4 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘(◡𝐺‘𝑥)) ≠ (◡𝐺‘𝑥) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
66	49, 65	bitrid 286	. . 3 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → ((◡𝐺‘𝑥) ≠ (𝐹‘(◡𝐺‘𝑥)) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
67	27, 48, 66	3bitrd 308	. 2 ⊢ (((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) ↔ 𝑥 ∈ (𝐺 “ dom (𝐹 ∖ I ))))
68	11, 17, 67	eqrdav 2762	1 ⊢ ((𝐹:𝐴⟶𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → dom (((𝐺 ∘ 𝐹) ∘ ◡𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900   ∖ cdif 4048   I cid 5552  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   “ cima 5661   ∘ ccom 5662   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by:  pmtrfconj  19731  psgnunilem1  19758