Theorem f1otrspeq 19391

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 6783	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹 Fn 𝐴)
2	1	ad2antrr 727	. 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 6783	. . 3 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺 Fn 𝐴)
4	3	ad2antlr 728	. 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 8578	. . . . . . 7 ⊢ 1o ∈ ω
6		simplrr 778	. . . . . . . 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 777	. . . . . . . . 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 8408	. . . . . . . . 9 ⊢ 2o = suc 1o
9	7, 8	breqtrdi 5141	. . . . . . . 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 5122	. . . . . . 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 484	. . . . . . 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 9099	. . . . . . 7 ⊢ ((1o ∈ ω ∧ dom (𝐺 ∖ I ) ≈ suc 1o ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o)
13	5, 10, 11, 12	mp3an2i 1469	. . . . . 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 8977	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∖ {𝑥}) ≈ 1o ↔ ∃!𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
15		eumo 2579	. . . . . . 7 ⊢ (∃!𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) → ∃*𝑦 𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
16	14, 15	sylbi 217	. . . . . 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 19387	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐹 ∖ I )) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥}))
19	18	ex 412	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → (𝑥 ∈ dom (𝐹 ∖ I ) → (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
20	19	ad2antrr 727	. . . . . . 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 2826	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ 𝑥 ∈ dom (𝐹 ∖ I )))
22	21	ad2antll 730	. . . . . . 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 4073	. . . . . . . . 9 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → (dom (𝐺 ∖ I ) ∖ {𝑥}) = (dom (𝐹 ∖ I ) ∖ {𝑥}))
24	23	eleq2d 2823	. . . . . . . 8 ⊢ (dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ) → ((𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐹 ∖ I ) ∖ {𝑥})))
25	24	ad2antll 730	. . . . . . 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 406	. . . . 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 19387	. . . . . 6 ⊢ ((𝐺:𝐴–1-1-onto→𝐴 ∧ 𝑥 ∈ dom (𝐺 ∖ I )) → (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}))
29	28	ad4ant24 755	. . . . 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 6855	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
31		fvex 6855	. . . . . . 7 ⊢ (𝐺‘𝑥) ∈ V
32	30, 31	pm3.2i 470	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ V ∧ (𝐺‘𝑥) ∈ V)
33		eleq1 2825	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐹‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
34		eleq1 2825	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑥) → (𝑦 ∈ (dom (𝐺 ∖ I ) ∖ {𝑥}) ↔ (𝐺‘𝑥) ∈ (dom (𝐺 ∖ I ) ∖ {𝑥})))
35	33, 34	moi 3678	. . . . . 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 1451	. . . . 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 837	. . . 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 716	. . 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 778	. . . . . . . 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 2823	. . . . . . 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 7133	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
42	2, 41	sylan 581	. . . . . . 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 279	. . . . . 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 2976	. . . . 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 477	. . . 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 7133	. . . . . . 7 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐺 ∖ I ) ↔ (𝐺‘𝑥) ≠ 𝑥))
47	4, 46	sylan 581	. . . . . 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 2976	. . . . 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 477	. . . 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 2775	. . 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 813	. 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 6988	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃*wmo 2538  ∃!weu 2569   ≠ wne 2933  Vcvv 3442   ∖ cdif 3900  {csn 4582   class class class wbr 5100   I cid 5526  dom cdm 5632  suc csuc 6327   Fn wfn 6495  –1-1-onto→wf1o 6499  ‘cfv 6500  ωcom 7818  1_oc1o 8400  2_oc2o 8401   ≈ cen 8892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-2o 8408  df-en 8896

This theorem is referenced by:  pmtrfb  19409  psgnunilem1  19437