Theorem f1omvdco3 18795

Description: If a point is moved by exactly one of two permutations, then it will be moved by their composite. (Contributed by Stefan O'Rear, 23-Aug-2015.)

Assertion

Ref	Expression
f1omvdco3	⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴 ∧ (𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I ))) → 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))

Proof of Theorem f1omvdco3

Step	Hyp	Ref	Expression
1		notbi 322	. . . . 5 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (¬ 𝑋 ∈ dom (𝐹 ∖ I ) ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I )))
2		disjsn 4613	. . . . . . 7 ⊢ ((dom (𝐹 ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom (𝐹 ∖ I ))
3		disj2 4358	. . . . . . 7 ⊢ ((dom (𝐹 ∖ I ) ∩ {𝑋}) = ∅ ↔ dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}))
4	2, 3	bitr3i 280	. . . . . 6 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∖ I ) ↔ dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}))
5		disjsn 4613	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I ))
6		disj2 4358	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∩ {𝑋}) = ∅ ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋}))
7	5, 6	bitr3i 280	. . . . . 6 ⊢ (¬ 𝑋 ∈ dom (𝐺 ∖ I ) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋}))
8	4, 7	bibi12i 343	. . . . 5 ⊢ ((¬ 𝑋 ∈ dom (𝐹 ∖ I ) ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
9	1, 8	bitri 278	. . . 4 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
10	9	notbii 323	. . 3 ⊢ (¬ (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ ¬ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
11		df-xor 1508	. . 3 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ ¬ (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )))
12		df-xor 1508	. . 3 ⊢ ((dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ⊻ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})) ↔ ¬ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
13	10, 11, 12	3bitr4i 306	. 2 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ⊻ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
14		f1omvdco2 18794	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴 ∧ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ⊻ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋}))) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}))
15		disj2 4358	. . . . 5 ⊢ ((dom ((𝐹 ∘ 𝐺) ∖ I ) ∩ {𝑋}) = ∅ ↔ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}))
16		disjsn 4613	. . . . 5 ⊢ ((dom ((𝐹 ∘ 𝐺) ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))
17	15, 16	bitr3i 280	. . . 4 ⊢ (dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}) ↔ ¬ 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))
18	17	con2bii 361	. . 3 ⊢ (𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ) ↔ ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}))
19	14, 18	sylibr 237	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴 ∧ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ⊻ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋}))) → 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))
20	13, 19	syl3an3b 1407	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴 ∧ (𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I ))) → 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ w3a 1089   ⊻ wxo 1507   = wceq 1543   ∈ wcel 2112  Vcvv 3398   ∖ cdif 3850   ∩ cin 3852   ⊆ wss 3853  ∅c0 4223  {csn 4527   I cid 5439  dom cdm 5536   ∘ ccom 5540  –1-1-onto→wf1o 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-xor 1508  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366

This theorem is referenced by:  psgnunilem5  18840