Theorem f1otrspeq 19055

Description: A transposition is characterized by the points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
f1otrspeq	⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → 𝐹 = 𝐺)

Proof of Theorem f1otrspeq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1ofn 6717	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
2	1	ad2antrr 723	. 2 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → 𝐹 Fn 𝐴)
3		f1ofn 6717	. . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺 Fn 𝐴)
4	3	ad2antlr 724	. 2 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → 𝐺 Fn 𝐴)
5		1onn 8470	. . . . . . 7 ⊢ 1o ∈ ω
6		simplrr 775	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))
7		simplrl 774	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → dom (𝐹 ∖ I ) ≈ 2o)
8		df-2o 8298	. . . . . . . . 9 ⊢ 2o = suc 1o
9	7, 8	breqtrdi 5115	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → dom (𝐹 ∖ I ) ≈ suc 1o)
10	6, 9	eqbrtrd 5096	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → dom (𝐺 ∖ I ) ≈ suc 1o)
11		simpr 485	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → 𝑥 ∈ dom (𝐺 ∖ I ))
12		dif1en 8945	. . . . . . 7 ⊢ ((1o ∈ ω ∧ dom (𝐺 ∖ I ) ≈ suc 1o ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o)
13	5, 10, 11, 12	mp3an2i 1465	. . . . . 6 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o)
14		euen1b 8817	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o ↔ ∃!𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
15		eumo 2578	. . . . . . 7 ⊢ (∃!𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) → ∃*𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
16	14, 15	sylbi 216	. . . . . 6 ⊢ ((dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o → ∃*𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
17	13, 16	syl 17	. . . . 5 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → ∃*𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
18		f1omvdmvd 19051	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
19	18	ex 413	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → (𝑥 ∈ dom (𝐹 ∖ I ) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
20	19	ad2antrr 723	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → (𝑥 ∈ dom (𝐹 ∖ I ) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
21		eleq2 2827	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ 𝑥 ∈ dom (𝐹 ∖ I )))
22	21	ad2antll 726	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ 𝑥 ∈ dom (𝐹 ∖ I )))
23		difeq1 4050	. . . . . . . . 9 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (dom (𝐺 ∖ I ) ∖ {𝑥}) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
24	23	eleq2d 2824	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → ((𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
25	24	ad2antll 726	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → ((𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
26	20, 22, 25	3imtr4d 294	. . . . . 6 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → (𝑥 ∈ dom (𝐺 ∖ I ) → (𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
27	26	imp 407	. . . . 5 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
28		f1omvdmvd 19051	. . . . . 6 ⊢ ((𝐺:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
29	28	ad4ant24 751	. . . . 5 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
30		fvex 6787	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
31		fvex 6787	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
32	30, 31	pm3.2i 471	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ V ∧ (𝐺‘𝑥) ∈ V)
33		eleq1 2826	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
34		eleq1 2826	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
35	33, 34	moi 3653	. . . . . 6 ⊢ ((((𝐹‘𝑥) ∈ V ∧ (𝐺‘𝑥) ∈ V) ∧ ∃*𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ∧ ((𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ∧ (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))) → (𝐹‘𝑥) = (𝐺‘𝑥))
36	32, 35	mp3an1 1447	. . . . 5 ⊢ ((∃*𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ∧ ((𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ∧ (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))) → (𝐹‘𝑥) = (𝐺‘𝑥))
37	17, 27, 29, 36	syl12anc 834	. . . 4 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐹‘𝑥) = (𝐺‘𝑥))
38	37	adantlr 712	. . 3 ⊢ (((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐹‘𝑥) = (𝐺‘𝑥))
39		simplrr 775	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) → dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))
40	39	eleq2d 2824	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ 𝑥 ∈ dom (𝐹 ∖ I )))
41		fnelnfp 7049	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
42	2, 41	sylan 580	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
43	40, 42	bitrd 278	. . . . . 6 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
44	43	necon2bbid 2987	. . . . 5 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑥 ↔ ¬ 𝑥 ∈ dom (𝐺 ∖ I )))
45	44	biimpar 478	. . . 4 ⊢ (((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐹‘𝑥) = 𝑥)
46		fnelnfp 7049	. . . . . . 7 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ (𝐺‘𝑥) ≠ 𝑥))
47	4, 46	sylan 580	. . . . . 6 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ (𝐺‘𝑥) ≠ 𝑥))
48	47	necon2bbid 2987	. . . . 5 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) → ((𝐺‘𝑥) = 𝑥 ↔ ¬ 𝑥 ∈ dom (𝐺 ∖ I )))
49	48	biimpar 478	. . . 4 ⊢ (((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐺‘𝑥) = 𝑥)
50	45, 49	eqtr4d 2781	. . 3 ⊢ (((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐹‘𝑥) = (𝐺‘𝑥))
51	38, 50	pm2.61dan 810	. 2 ⊢ ((((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))
52	2, 4, 51	eqfnfvd 6912	1 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2o ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → 𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃*wmo 2538  ∃!weu 2568   ≠ wne 2943  Vcvv 3432   ∖ cdif 3884  {csn 4561   class class class wbr 5074   I cid 5488  dom cdm 5589  suc csuc 6268   Fn wfn 6428  –1-1-onto→wf1o 6432  ‘cfv 6433  ωcom 7712  1_oc1o 8290  2_oc2o 8291   ≈ cen 8730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-2o 8298  df-en 8734

This theorem is referenced by:  pmtrfb  19073  psgnunilem1  19101