Theorem f1omvdco2 18578

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 1507	. . 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 6120	. . . . . . . . . . . 12 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺))
3		f1ococnv1 6645	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
4	3	coeq1d 5734	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
5		f1of 6617	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → 𝐺:𝐴⟶𝐴)
6		fcoi2 6555	. . . . . . . . . . . . . 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 2878	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺)
9	2, 8	syl5eqr 2872	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺)
10	9	difeq1d 4100	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) = (𝐺 ∖ I ))
11	10	dmeqd 5776	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) = dom (𝐺 ∖ I ))
12	11	adantr 483	. . . . . . . 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 18575	. . . . . . . . 9 ⊢ dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) ⊆ (dom (◡𝐹 ∖ I ) ∪ dom ((𝐹 ∘ 𝐺) ∖ I ))
14		f1omvdcnv 18574	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))
15	14	ad2antrr 724	. . . . . . . . . . 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 769	. . . . . . . . . . 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 4007	. . . . . . . . . 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 771	. . . . . . . . . 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 4164	. . . . . . . . 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 3980	. . . . . . . 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 4008	. . . . . . 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 459	. . . . . 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 456	. . . 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 6120	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺))
26		f1ococnv2 6643	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝐴))
27	26	coeq2d 5735	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ 𝐴)))
28		f1of 6617	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴⟶𝐴)
29		fcoi1 6554	. . . . . . . . . . . . . . 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 2880	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = 𝐹)
32	25, 31	syl5eq 2870	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = 𝐹)
33	32	difeq1d 4100	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) = (𝐹 ∖ I ))
34	33	dmeqd 5776	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→𝐴 ∧ 𝐺:𝐴–1-1-onto→𝐴) → dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) = dom (𝐹 ∖ I ))
35	34	adantr 483	. . . . . . . . 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 18575	. . . . . . . . . 10 ⊢ dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) ⊆ (dom ((𝐹 ∘ 𝐺) ∖ I ) ∪ dom (◡𝐺 ∖ I ))
37		simprr 771	. . . . . . . . . . 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 18574	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴–1-1-onto→𝐴 → dom (◡𝐺 ∖ I ) = dom (𝐺 ∖ I ))
39	38	ad2antlr 725	. . . . . . . . . . . 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 769	. . . . . . . . . . . 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 4007	. . . . . . . . . . 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 4164	. . . . . . . . . 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 3980	. . . . . . . . 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 4008	. . . . . . . 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 459	. . . . . . 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 456	. . . . 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 468	. . . 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 855	. . 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 244	. 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 1113	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 398   ∨ wo 843   ∧ w3a 1083   ⊻ wxo 1501   = wceq 1537   ∖ cdif 3935   ∪ cun 3936   ⊆ wss 3938   I cid 5461  ^◡ccnv 5556  dom cdm 5557   ↾ cres 5559   ∘ ccom 5561  ⟶wf 6353  –1-1-onto→wf1o 6356

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-xor 1502  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365

This theorem is referenced by:  f1omvdco3  18579