Theorem f1omvdco3 19447

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 318	. . . . 5 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (¬ 𝑋 ∈ dom (𝐹 ∖ I ) ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I )))
2		disjsn 4720	. . . . . . 7 ⊢ ((dom (𝐹 ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom (𝐹 ∖ I ))
3		disj2 4462	. . . . . . 7 ⊢ ((dom (𝐹 ∖ I ) ∩ {𝑋}) = ∅ ↔ dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}))
4	2, 3	bitr3i 276	. . . . . 6 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∖ I ) ↔ dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}))
5		disjsn 4720	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I ))
6		disj2 4462	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∩ {𝑋}) = ∅ ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋}))
7	5, 6	bitr3i 276	. . . . . 6 ⊢ (¬ 𝑋 ∈ dom (𝐺 ∖ I ) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋}))
8	4, 7	bibi12i 338	. . . . 5 ⊢ ((¬ 𝑋 ∈ dom (𝐹 ∖ I ) ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
9	1, 8	bitri 274	. . . 4 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
10	9	notbii 319	. . 3 ⊢ (¬ (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ ¬ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ↔ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
11		df-xor 1506	. . 3 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ ¬ (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )))
12		df-xor 1506	. . 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 302	. 2 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ (dom (𝐹 ∖ I ) ⊆ (V ∖ {𝑋}) ⊻ dom (𝐺 ∖ I ) ⊆ (V ∖ {𝑋})))
14		f1omvdco2 19446	. . 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 4462	. . . . 5 ⊢ ((dom ((𝐹 ∘ 𝐺) ∖ I ) ∩ {𝑋}) = ∅ ↔ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}))
16		disjsn 4720	. . . . 5 ⊢ ((dom ((𝐹 ∘ 𝐺) ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))
17	15, 16	bitr3i 276	. . . 4 ⊢ (dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}) ↔ ¬ 𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ))
18	17	con2bii 356	. . 3 ⊢ (𝑋 ∈ dom ((𝐹 ∘ 𝐺) ∖ I ) ↔ ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}))
19	14, 18	sylibr 233	. 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 1402	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 205   ∧ w3a 1084   ⊻ wxo 1505   = wceq 1534   ∈ wcel 2099  Vcvv 3462   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  ∅c0 4325  {csn 4633   I cid 5579  dom cdm 5682   ∘ ccom 5686  –1-1-onto→wf1o 6553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-xor 1506  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562

This theorem is referenced by:  psgnunilem5  19492