Theorem f1omvdco2 19713

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⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴 ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋)) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)

Proof of Theorem f1omvdco2

Step	Hyp	Ref	Expression
1		excxor 1546	. . 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 6290	. . . . . . . . . . . 12 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺))
3		f1ococnv1 6885	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐴⟶1-1onto𝐴 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
4	3	coeq1d 5850	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴⟶1-1onto𝐴 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
5		f1of 6854	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴⟶1-1onto𝐴 → 𝐺:𝐴⟶𝐴)
6		fcoi2 6787	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
7	5, 6	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴⟶1-1onto𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
8	4, 7	sylan9eq 2818	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺)
9	2, 8	eqtr3id 2812	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺)
10	9	difeq1d 4065	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) = (𝐺 ∖ I ))
11	10	dmeqd 5899	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) = dom (𝐺 ∖ I ))
12	11	adantr 486	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) = dom (𝐺 ∖ I ))
13		mvdco 19710	. . . . . . . . 9 ⊢ dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) ⊆ (dom (◡𝐹 ∖ I ) ∪ dom ((𝐹 ∘ 𝐺) ∖ I ))
14		f1omvdcnv 19709	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶1-1onto𝐴 → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))
15	14	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (◡𝐹 ∖ I ) = dom (𝐹 ∖ I ))
16		simprl 783	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (𝐹 ∖ I ) ⊆ 𝑋)
17	15, 16	eqsstrd 3947	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (◡𝐹 ∖ I ) ⊆ 𝑋)
18		simprr 785	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)
19	17, 18	unssd 4193	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → (dom (◡𝐹 ∖ I ) ∪ dom ((𝐹 ∘ 𝐺) ∖ I )) ⊆ 𝑋)
20	13, 19	sstrid 3924	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom ((◡𝐹 ∘ (𝐹 ∘ 𝐺)) ∖ I ) ⊆ 𝑋)
21	12, 20	eqsstrrd 3948	. . . . . . 7 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (𝐺 ∖ I ) ⊆ 𝑋)
22	21	expr 462	. . . . . 6 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ dom (𝐹 ∖ I ) ⊆ 𝑋) → (dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋 → dom (𝐺 ∖ I ) ⊆ 𝑋))
23	22	con3d 153	. . . . 5 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ dom (𝐹 ∖ I ) ⊆ 𝑋) → (¬ dom (𝐺 ∖ I ) ⊆ 𝑋 → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
24	23	expimpd 459	. . . 4 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ((dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ ¬ dom (𝐺 ∖ I ) ⊆ 𝑋) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
25		coass 6290	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ∘ (𝐺 ∘ ◡𝐺))
26		f1ococnv2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐴⟶1-1onto𝐴 → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝐴))
27	26	coeq2d 5851	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴⟶1-1onto𝐴 → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = (𝐹 ∘ ( I ↾ 𝐴)))
28		f1of 6854	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐴⟶1-1onto𝐴 → 𝐹:𝐴⟶𝐴)
29		fcoi1 6786	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐴⟶𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
30	28, 29	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐴⟶1-1onto𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
31	27, 30	sylan9eqr 2820	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → (𝐹 ∘ (𝐺 ∘ ◡𝐺)) = 𝐹)
32	25, 31	eqtrid 2810	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = 𝐹)
33	32	difeq1d 4065	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) = (𝐹 ∖ I ))
34	33	dmeqd 5899	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) = dom (𝐹 ∖ I ))
35	34	adantr 486	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) = dom (𝐹 ∖ I ))
36		mvdco 19710	. . . . . . . . . 10 ⊢ dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) ⊆ (dom ((𝐹 ∘ 𝐺) ∖ I ) ∪ dom (◡𝐺 ∖ I ))
37		simprr 785	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)
38		f1omvdcnv 19709	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴⟶1-1onto𝐴 → dom (◡𝐺 ∖ I ) = dom (𝐺 ∖ I ))
39	38	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (◡𝐺 ∖ I ) = dom (𝐺 ∖ I ))
40		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (𝐺 ∖ I ) ⊆ 𝑋)
41	39, 40	eqsstrd 3947	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (◡𝐺 ∖ I ) ⊆ 𝑋)
42	37, 41	unssd 4193	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → (dom ((𝐹 ∘ 𝐺) ∖ I ) ∪ dom (◡𝐺 ∖ I )) ⊆ 𝑋)
43	36, 42	sstrid 3924	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (((𝐹 ∘ 𝐺) ∘ ◡𝐺) ∖ I ) ⊆ 𝑋)
44	35, 43	eqsstrrd 3948	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ (dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)) → dom (𝐹 ∖ I ) ⊆ 𝑋)
45	44	expr 462	. . . . . . 7 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ dom (𝐺 ∖ I ) ⊆ 𝑋) → (dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋 → dom (𝐹 ∖ I ) ⊆ 𝑋))
46	45	con3d 153	. . . . . 6 ⊢ (((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) ∧ dom (𝐺 ∖ I ) ⊆ 𝑋) → (¬ dom (𝐹 ∖ I ) ⊆ 𝑋 → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
47	46	expimpd 459	. . . . 5 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ((dom (𝐺 ∖ I ) ⊆ 𝑋 ∧ ¬ dom (𝐹 ∖ I ) ⊆ 𝑋) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
48	47	ancomsd 471	. . . 4 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ((¬ dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom (𝐺 ∖ I ) ⊆ 𝑋) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
49	24, 48	jaod 873	. . 3 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → (((dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ ¬ dom (𝐺 ∖ I ) ⊆ 𝑋) ∨ (¬ dom (𝐹 ∖ I ) ⊆ 𝑋 ∧ dom (𝐺 ∖ I ) ⊆ 𝑋)) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
50	1, 49	biimtrid 245	. 2 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴) → ((dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋))
51	50	3impia 1135	1 ⊢ ((𝐹:𝐴⟶1-1onto𝐴 ∧ 𝐺:𝐴⟶1-1onto𝐴 ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋)) → ¬ dom ((𝐹 ∘ 𝐺) ∖ I ) ⊆ 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   ⊻ wxo 1541   = wceq 1570   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152   I cid 5552  ^◡ccnv 5657  dom cdm 5658   ↾ cres 5660   ∘ ccom 5662  ⟶wf 6562  ⟶1-1ontowf1o 6565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by:  f1omvdco3  19714