Theorem fresf1o 29433

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 5918	. . . . . . . 8 ⊢ (Fun (◡𝐹 ↾ 𝐶) ↔ (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
2	1	biimpi 206	. . . . . . 7 ⊢ (Fun (◡𝐹 ↾ 𝐶) → (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
3	2	3ad2ant3 1084	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
4		simp2 1062	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → 𝐶 ⊆ ran 𝐹)
5		df-rn 5125	. . . . . . . . 9 ⊢ ran 𝐹 = dom ◡𝐹
6	4, 5	syl6sseq 3651	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → 𝐶 ⊆ dom ◡𝐹)
7		ssdmres 5420	. . . . . . . 8 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
8	6, 7	sylib 208	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
9	8	fneq2d 5982	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ((◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶) ↔ (◡𝐹 ↾ 𝐶) Fn 𝐶))
10	3, 9	mpbid 222	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶) Fn 𝐶)
11		simp1 1061	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun 𝐹)
12		funres 5929	. . . . . . 7 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ (◡𝐹 “ 𝐶)))
13	11, 12	syl 17	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun (𝐹 ↾ (◡𝐹 “ 𝐶)))
14		funcnvres2 5969	. . . . . . . 8 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
15	11, 14	syl 17	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)))
16	15	funeqd 5910	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (Fun ◡(◡𝐹 ↾ 𝐶) ↔ Fun (𝐹 ↾ (◡𝐹 “ 𝐶))))
17	13, 16	mpbird 247	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun ◡(◡𝐹 ↾ 𝐶))
18		df-ima 5127	. . . . . . 7 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
19	18	eqcomi 2631	. . . . . 6 ⊢ ran (◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)
20	19	a1i 11	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ran (◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶))
21	10, 17, 20	3jca 1242	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ((◡𝐹 ↾ 𝐶) Fn 𝐶 ∧ Fun ◡(◡𝐹 ↾ 𝐶) ∧ ran (◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)))
22		dff1o2 6142	. . . 4 ⊢ ((◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶) ↔ ((◡𝐹 ↾ 𝐶) Fn 𝐶 ∧ Fun ◡(◡𝐹 ↾ 𝐶) ∧ ran (◡𝐹 ↾ 𝐶) = (◡𝐹 “ 𝐶)))
23	21, 22	sylibr 224	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶))
24		f1ocnv 6149	. . 3 ⊢ ((◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)
25	23, 24	syl 17	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)
26		f1oeq1 6127	. . 3 ⊢ (◡(◡𝐹 ↾ 𝐶) = (𝐹 ↾ (◡𝐹 “ 𝐶)) → (◡(◡𝐹 ↾ 𝐶):(◡𝐹 “ 𝐶)–1-1-onto→𝐶 ↔ (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶))
27	11, 14, 26	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))
28	25, 27	mpbid 222	1 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ⊆ wss 3574  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Fun wfun 5882   Fn wfn 5883  –1-1-onto→wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  carsggect  30380