Theorem f1otrspeq 19600

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 6821	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
2	1	ad2antrr 739	. 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 6821	. . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺 Fn 𝐴)
4	3	ad2antlr 740	. 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 8635	. . . . . . 7 ⊢ 1o ∈ ω
6		simplrr 790	. . . . . . . 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 789	. . . . . . . . 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 8463	. . . . . . . . 9 ⊢ 2o = suc 1o
9	7, 8	breqtrdi 5146	. . . . . . . 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 5127	. . . . . . 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 490	. . . . . . 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 9164	. . . . . . 7 ⊢ ((1o ∈ ω ∧ dom (𝐺 ∖ I ) ≈ suc 1o ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o)
13	5, 10, 11, 12	mp3an2i 1495	. . . . . 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 9041	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o ↔ ∃!𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
15		eumo 2603	. . . . . . 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 19596	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
19	18	ex 418	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → (𝑥 ∈ dom (𝐹 ∖ I ) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
20	19	ad2antrr 739	. . . . . . 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 2849	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ 𝑥 ∈ dom (𝐹 ∖ I )))
22	21	ad2antll 742	. . . . . . 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 4067	. . . . . . . . 9 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (dom (𝐺 ∖ I ) ∖ {𝑥}) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
24	23	eleq2d 2846	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → ((𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
25	24	ad2antll 742	. . . . . . 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 412	. . . . 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 19596	. . . . . 6 ⊢ ((𝐺:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
29	28	ad4ant24 767	. . . . 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 6894	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
31		fvex 6894	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
32	30, 31	pm3.2i 476	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ V ∧ (𝐺‘𝑥) ∈ V)
33		eleq1 2848	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
34		eleq1 2848	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
35	33, 34	moi 3676	. . . . . 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 1477	. . . . 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 850	. . . 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 728	. . 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 790	. . . . . . . 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 2846	. . . . . . 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 7178	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
42	2, 41	sylan 592	. . . . . . 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 2998	. . . . 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 483	. . . 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 7178	. . . . . . 7 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ (𝐺‘𝑥) ≠ 𝑥))
47	4, 46	sylan 592	. . . . . 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 2998	. . . . 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 483	. . . 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 2798	. . 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 825	. 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 7028	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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃*wmo 2562  ∃!weu 2593   ≠ wne 2955  Vcvv 3450   ∖ cdif 3896  {csn 4584   class class class wbr 5103   I cid 5549  dom cdm 5655  suc csuc 6361   Fn wfn 6530  –1-1-onto→wf1o 6534  ‘cfv 6535  ωcom 7868  1_oc1o 8455  2_oc2o 8456   ≈ cen 8956

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-om 7869  df-1o 8462  df-2o 8463  df-en 8960

This theorem is used by:  pmtrfb  19618  psgnunilem1  19646