Theorem fresf1o 32563

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 6554	. . . . . . 7 ⊢ (Fun (◡𝐹 ↾ 𝐶) ↔ (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
2	1	biimpi 216	. . . . . 6 ⊢ (Fun (◡𝐹 ↾ 𝐶) → (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
3	2	3ad2ant3 1135	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
4		simp2 1137	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → 𝐶 ⊆ ran 𝐹)
5		df-rn 5657	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
6	4, 5	sseqtrdi 3995	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → 𝐶 ⊆ dom ◡𝐹)
7		ssdmres 5992	. . . . . . 7 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
8	6, 7	sylib 218	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
9	8	fneq2d 6620	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ((◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶) ↔ (◡𝐹 ↾ 𝐶) Fn 𝐶))
10	3, 9	mpbid 232	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶) Fn 𝐶)
11		simp1 1136	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun 𝐹)
12	11	funresd 6567	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun (𝐹 ↾ (◡𝐹 “ 𝐶)))
13		funcnvres2 6604	. . . . . . 7 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
14	11, 13	syl 17	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
15	14	funeqd 6546	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (Fun ◡(◡𝐹 ↾ 𝐶) ↔ Fun (𝐹 ↾ (◡𝐹 “ 𝐶))))
16	12, 15	mpbird 257	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun ◡(◡𝐹 ↾ 𝐶))
17		df-ima 5659	. . . . . 6 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
18	17	eqcomi 2739	. . . . 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 6812	. . . 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 1344	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶))
22		f1ocnv 6819	. . 3 ⊢ ((◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)
23	21, 22	syl 17	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)
24		f1oeq1 6795	. . 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 18	. 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 232	1 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ⊆ wss 3922  ^◡ccnv 5645  dom cdm 5646  ran crn 5647   ↾ cres 5648   “ cima 5649  Fun wfun 6513   Fn wfn 6514  –1-1-onto→wf1o 6518

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-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526

This theorem is referenced by:  carsggect  34317