Theorem f1omvdco3 19355

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 4671	. . . . . . 7 ⊢ ((dom (𝐹 ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom (𝐹 ∖ I ))
3		disj2 4417	. . . . . . 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 4671	. . . . . . 7 ⊢ ((dom (𝐺 ∖ I ) ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ dom (𝐺 ∖ I ))
6		disj2 4417	. . . . . . 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 1512	. . 3 ⊢ ((𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I )) ↔ ¬ (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ dom (𝐺 ∖ I )))
12		df-xor 1512	. . 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 19354	. . 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 4417	. . . . 5 ⊢ ((dom ((𝐹 ∘ 𝐺) ∖ I ) ∩ {𝑋}) = ∅ ↔ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ (V ∖ {𝑋}))
16		disjsn 4671	. . . . 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 1511   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4585   I cid 5525  dom cdm 5631   ∘ ccom 5635  –1-1-onto→wf1o 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507

This theorem is referenced by:  psgnunilem5  19400