Theorem f1omvdco2 18643

Description: If exactly one of two permutations is limited to a set of points, then the composition will not be. (Contributed by Stefan O'Rear, 23-Aug-2015.)

Assertion

Ref	Expression
f1omvdco2	⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴 ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋)) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)

Proof of Theorem f1omvdco2

Step	Hyp	Ref	Expression
1		excxor 1508	. . 3 ⊢ ((dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋) ↔ ((dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ ¬ dom (𝐺 ∖ I ) ⊆ 𝑋) ∨ (¬ dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom (𝐺 ∖ I ) ⊆ 𝑋)))
2		coass 6095	. . . . . . . . . . . 12 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺))
3		f1ococnv1 6630	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
4	3	coeq1d 5701	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
5		f1of 6602	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺:𝐴⟶𝐴)
6		fcoi2 6538	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
7	5, 6	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
8	4, 7	sylan9eq 2813	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺)
9	2, 8	syl5eqr 2807	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺)
10	9	difeq1d 4027	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) = (𝐺 ∖ I ))
11	10	dmeqd 5745	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) = dom (𝐺 ∖ I ))
12	11	adantr 484	. . . . . . . 8 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) = dom (𝐺 ∖ I ))
13		mvdco 18640	. . . . . . . . 9 ⊢ dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) ⊆ (dom (◡𝐹 ∖ I ) ∪ dom ((𝐹 ∘ 𝐺) ∖ I ))
14		f1omvdcnv 18639	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))
15	14	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))
16		simprl 770	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (𝐹 ∖ I ) ⊆ 𝑋)
17	15, 16	eqsstrd 3930	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (◡𝐹 ∖ I ) ⊆ 𝑋)
18		simprr 772	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)
19	17, 18	unssd 4091	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → (dom (◡𝐹 ∖ I ) ∪ dom ((𝐹 ∘ 𝐺) ∖ I )) ⊆ 𝑋)
20	13, 19	sstrid 3903	. . . . . . . 8 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) ⊆ 𝑋)
21	12, 20	eqsstrrd 3931	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (𝐺 ∖ I ) ⊆ 𝑋)
22	21	expr 460	. . . . . 6 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ dom (𝐹 ∖ I ) ⊆ 𝑋) → (dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋 → dom (𝐺 ∖ I ) ⊆ 𝑋))
23	22	con3d 155	. . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ dom (𝐹 ∖ I ) ⊆ 𝑋) → (¬ dom (𝐺 ∖ I ) ⊆ 𝑋 → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
24	23	expimpd 457	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ ¬ dom (𝐺 ∖ I ) ⊆ 𝑋) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
25		coass 6095	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺))
26		f1ococnv2 6628	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝐴))
27	26	coeq2d 5702	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ 𝐴)))
28		f1of 6602	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴)
29		fcoi1 6537	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐴⟶𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
30	28, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
31	27, 30	sylan9eqr 2815	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = 𝐹)
32	25, 31	syl5eq 2805	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = 𝐹)
33	32	difeq1d 4027	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) = (𝐹 ∖ I ))
34	33	dmeqd 5745	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) = dom (𝐹 ∖ I ))
35	34	adantr 484	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) = dom (𝐹 ∖ I ))
36		mvdco 18640	. . . . . . . . . 10 ⊢ dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) ⊆ (dom ((𝐹 ∘ 𝐺) ∖ I ) ∪ dom (◡𝐺 ∖ I ))
37		simprr 772	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)
38		f1omvdcnv 18639	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → dom (◡𝐺 ∖ I ) = dom (𝐺 ∖ I ))
39	38	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (◡𝐺 ∖ I ) = dom (𝐺 ∖ I ))
40		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (𝐺 ∖ I ) ⊆ 𝑋)
41	39, 40	eqsstrd 3930	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (◡𝐺 ∖ I ) ⊆ 𝑋)
42	37, 41	unssd 4091	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → (dom ((𝐹 ∘ 𝐺) ∖ I ) ∪ dom (◡𝐺 ∖ I )) ⊆ 𝑋)
43	36, 42	sstrid 3903	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) ⊆ 𝑋)
44	35, 43	eqsstrrd 3931	. . . . . . . 8 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (𝐹 ∖ I ) ⊆ 𝑋)
45	44	expr 460	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ dom (𝐺 ∖ I ) ⊆ 𝑋) → (dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋 → dom (𝐹 ∖ I ) ⊆ 𝑋))
46	45	con3d 155	. . . . . 6 ⊢ (((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) ∧ dom (𝐺 ∖ I ) ⊆ 𝑋) → (¬ dom (𝐹 ∖ I ) ⊆ 𝑋 → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
47	46	expimpd 457	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ ¬ dom (𝐹 ∖ I ) ⊆ 𝑋) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
48	47	ancomsd 469	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((¬ dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom (𝐺 ∖ I ) ⊆ 𝑋) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
49	24, 48	jaod 856	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (((dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ ¬ dom (𝐺 ∖ I ) ⊆ 𝑋) ∨ (¬ dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom (𝐺 ∖ I ) ⊆ 𝑋)) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
50	1, 49	syl5bi 245	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
51	50	3impia 1114	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴 ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋)) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   ⊻ wxo 1502   = wceq 1538   ∖ cdif 3855   ∪ cun 3856   ⊆ wss 3858   I cid 5429  ^◡ccnv 5523  dom cdm 5524   ↾ cres 5526   ∘ ccom 5528  ⟶wf 6331  –1-1-onto→wf1o 6334

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-xor 1503  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343

This theorem is referenced by:  f1omvdco3  18644