Theorem f1otrspeq 19489

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 6809	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
2	1	ad2antrr 736	. 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 6809	. . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺 Fn 𝐴)
4	3	ad2antlr 737	. 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 8612	. . . . . . 7 ⊢ 1o ∈ ω
6		simplrr 787	. . . . . . . 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 786	. . . . . . . . 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 8440	. . . . . . . . 9 ⊢ 2o = suc 1o
9	7, 8	breqtrdi 5143	. . . . . . . 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 5124	. . . . . . 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 488	. . . . . . 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 9133	. . . . . . 7 ⊢ ((1o ∈ ω ∧ dom (𝐺 ∖ I ) ≈ suc 1o ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o)
13	5, 10, 11, 12	mp3an2i 1489	. . . . . 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 9011	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o ↔ ∃!𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
15		eumo 2607	. . . . . . 7 ⊢ (∃!𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) → ∃*𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
16	14, 15	sylbi 219	. . . . . 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 19485	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
19	18	ex 416	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → (𝑥 ∈ dom (𝐹 ∖ I ) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
20	19	ad2antrr 736	. . . . . . 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 2853	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ 𝑥 ∈ dom (𝐹 ∖ I )))
22	21	ad2antll 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 (𝐺 ∖ I ) ↔ 𝑥 ∈ dom (𝐹 ∖ I )))
23		difeq1 4075	. . . . . . . . 9 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (dom (𝐺 ∖ I ) ∖ {𝑥}) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
24	23	eleq2d 2850	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → ((𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
25	24	ad2antll 739	. . . . . . 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 296	. . . . . 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 410	. . . . 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 19485	. . . . . 6 ⊢ ((𝐺:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
29	28	ad4ant24 764	. . . . 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 6882	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
31		fvex 6882	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
32	30, 31	pm3.2i 474	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ V ∧ (𝐺‘𝑥) ∈ V)
33		eleq1 2852	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
34		eleq1 2852	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
35	33, 34	moi 3683	. . . . . 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 1471	. . . . 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 847	. . . 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 725	. . 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 787	. . . . . . . 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 2850	. . . . . . 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 7163	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
42	2, 41	sylan 589	. . . . . . 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 281	. . . . . 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 3002	. . . . 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 481	. . . 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 7163	. . . . . . 7 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ (𝐺‘𝑥) ≠ 𝑥))
47	4, 46	sylan 589	. . . . . 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 3002	. . . . 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 481	. . . 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 2802	. . 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 822	. 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 7016	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 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∃*wmo 2566  ∃!weu 2597   ≠ wne 2959  Vcvv 3456   ∖ cdif 3903  {csn 4584   class class class wbr 5102   I cid 5543  dom cdm 5649  suc csuc 6350   Fn wfn 6518  –1-1-onto→wf1o 6522  ‘cfv 6523  ωcom 7848  1_oc1o 8432  2_oc2o 8433   ≈ cen 8926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-om 7849  df-1o 8439  df-2o 8440  df-en 8930

This theorem is referenced by:  pmtrfb  19507  psgnunilem1  19535