Theorem f1otrspeq 19517

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 6822	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
2	1	ad2antrr 738	. 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 6822	. . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺 Fn 𝐴)
4	3	ad2antlr 739	. 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 8626	. . . . . . 7 ⊢ 1o ∈ ω
6		simplrr 789	. . . . . . . 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 788	. . . . . . . . 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 8454	. . . . . . . . 9 ⊢ 2o = suc 1o
9	7, 8	breqtrdi 5156	. . . . . . . 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 5137	. . . . . . 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 489	. . . . . . 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		dif1ennn 9147	. . . . . . 7 ⊢ ((1o ∈ ω ∧ dom (𝐺 ∖ I ) ≈ suc 1o ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o)
13	5, 10, 11, 12	mp3an2i 1492	. . . . . 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 9025	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o ↔ ∃!𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
15		eumo 2612	. . . . . . 7 ⊢ (∃!𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) → ∃*𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
16	14, 15	sylbi 220	. . . . . 6 ⊢ ((dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o → ∃*𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
17	13, 16	syl 18	. . . . 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 19513	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
19	18	ex 417	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → (𝑥 ∈ dom (𝐹 ∖ I ) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
20	19	ad2antrr 738	. . . . . . 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 2858	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ 𝑥 ∈ dom (𝐹 ∖ I )))
22	21	ad2antll 741	. . . . . . 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 4082	. . . . . . . . 9 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (dom (𝐺 ∖ I ) ∖ {𝑥}) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
24	23	eleq2d 2855	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → ((𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
25	24	ad2antll 741	. . . . . . 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 297	. . . . . 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 411	. . . . 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 19513	. . . . . 6 ⊢ ((𝐺:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
29	28	ad4ant24 766	. . . . 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 6895	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
31		fvex 6895	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
32	30, 31	pm3.2i 475	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ V ∧ (𝐺‘𝑥) ∈ V)
33		eleq1 2857	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
34		eleq1 2857	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
35	33, 34	moi 3690	. . . . . 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 1474	. . . . 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 849	. . . 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 727	. . 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 789	. . . . . . . 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 2855	. . . . . . 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 7176	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
42	2, 41	sylan 591	. . . . . . 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 282	. . . . . 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 3007	. . . . 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 482	. . . 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 7176	. . . . . . 7 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ (𝐺‘𝑥) ≠ 𝑥))
47	4, 46	sylan 591	. . . . . 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 3007	. . . . 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 482	. . . 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 2807	. . 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 824	. 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 7029	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 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃*wmo 2571  ∃!weu 2602   ≠ wne 2964  Vcvv 3463   ∖ cdif 3910  {csn 4594   class class class wbr 5113   I cid 5556  dom cdm 5662  suc csuc 6363   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537  ωcom 7862  1_oc1o 8446  2_oc2o 8447   ≈ cen 8940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7863  df-1o 8453  df-2o 8454  df-en 8944

This theorem is referenced by:  pmtrfb  19535  psgnunilem1  19563