Theorem f1omvdco3 19378

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 319	. . . . 5 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (¬ 𝑋 ∈ dom (𝐹 ∖ I ) ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I )))
2		disjsn 4668	. . . . . . 7 ⊢ ((dom (𝐹 ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom (𝐹 ∖ I ))
3		disj2 4410	. . . . . . 7 ⊢ ((dom (𝐹 ∖ I ) ∩ {𝑋}) = ∅ ↔ dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}))
4	2, 3	bitr3i 277	. . . . . 6 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∖ I ) ↔ dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}))
5		disjsn 4668	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I ))
6		disj2 4410	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∩ {𝑋}) = ∅ ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋}))
7	5, 6	bitr3i 277	. . . . . 6 ⊢ (¬ 𝑋 ∈ dom (𝐺 ∖ I ) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋}))
8	4, 7	bibi12i 339	. . . . 5 ⊢ ((¬ 𝑋 ∈ dom (𝐹 ∖ I ) ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
9	1, 8	bitri 275	. . . 4 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
10	9	notbii 320	. . 3 ⊢ (¬ (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ ¬ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
11		df-xor 1513	. . 3 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ ¬ (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )))
12		df-xor 1513	. . 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 303	. 2 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ⊻ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
14		f1omvdco2 19377	. . 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 4410	. . . . 5 ⊢ ((dom ((𝐹 ∘ 𝐺) ∖ I ) ∩ {𝑋}) = ∅ ↔ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}))
16		disjsn 4668	. . . . 5 ⊢ ((dom ((𝐹 ∘ 𝐺) ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))
17	15, 16	bitr3i 277	. . . 4 ⊢ (dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}) ↔ ¬ 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))
18	17	con2bii 357	. . 3 ⊢ (𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ) ↔ ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}))
19	14, 18	sylibr 234	. 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 206   ∧ w3a 1086   ⊻ wxo 1512   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  {csn 4580   I cid 5518  dom cdm 5624   ∘ ccom 5628  –1-1-onto→wf1o 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-xor 1513  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  psgnunilem5  19423