Theorem fresf1o 33047

Description: Conditions for a restriction to be a one-to-one onto function. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Assertion

Ref	Expression
fresf1o	⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)

Proof of Theorem fresf1o

Step	Hyp	Ref	Expression
1		funfn 6570	. . . . . . 7 ⊢ (Fun (◡𝐹 ↾ 𝐶) ↔ (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
2	1	biimpi 219	. . . . . 6 ⊢ (Fun (◡𝐹 ↾ 𝐶) → (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
3	2	3ad2ant3 1153	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
4		simp2 1155	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → 𝐶 ⊆ ran 𝐹)
5		df-rn 5674	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
6	4, 5	sseqtrdi 3978	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → 𝐶 ⊆ dom ◡𝐹)
7		ssdmres 6014	. . . . . . 7 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
8	6, 7	sylib 221	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
9	8	fneq2d 6633	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ((◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶) ↔ (◡𝐹 ↾ 𝐶) Fn 𝐶))
10	3, 9	mpbid 235	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶) Fn 𝐶)
11		simp1 1154	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun 𝐹)
12	11	funresd 6583	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun (𝐹 ↾ (◡𝐹 “ 𝐶)))
13		funcnvres2 6620	. . . . . . 7 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
14	11, 13	syl 18	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
15	14	funeqd 6562	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (Fun ◡(◡𝐹 ↾ 𝐶) ↔ Fun (𝐹 ↾ (◡𝐹 “ 𝐶))))
16	12, 15	mpbird 260	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun ◡(◡𝐹 ↾ 𝐶))
17		df-ima 5676	. . . . . 6 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
18	17	eqcomi 2774	. . . . 5 ⊢ ran (◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)
19	18	a1i 11	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ran (◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶))
20		dff1o2 6830	. . . 4 ⊢ ((◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶) ↔ ((◡𝐹 ↾ 𝐶) Fn 𝐶 ∧ Fun ◡(◡𝐹 ↾ 𝐶) ∧ ran (◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)))
21	10, 16, 19, 20	syl3anbrc 1362	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶))
22		f1ocnv 6837	. . 3 ⊢ ((◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)
23	21, 22	syl 18	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)
24		f1oeq1 6812	. . 3 ⊢ (◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)) → (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–1-1-onto→𝐶 ↔ (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶))
25	11, 13, 24	3syl 19	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–1-1-onto→𝐶 ↔ (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶))
26	23, 25	mpbid 235	1 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ⊆ wss 3906  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666  Fun wfun 6534   Fn wfn 6535  –1-1-onto→wf1o 6539

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547

This theorem is used by:  carsggect  34773