Theorem fresf1o 31850

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 6578	. . . . . . 7 ⊢ (Fun (◡𝐹 ↾ 𝐶) ↔ (◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶))
2	1	biimpi 215	. . . . . 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 5687	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
6	4, 5	sseqtrdi 4032	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → 𝐶 ⊆ dom ◡𝐹)
7		ssdmres 6004	. . . . . . 7 ⊢ (𝐶 ⊆ dom ◡𝐹 ↔ dom (◡𝐹 ↾ 𝐶) = 𝐶)
8	6, 7	sylib 217	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → dom (◡𝐹 ↾ 𝐶) = 𝐶)
9	8	fneq2d 6643	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → ((◡𝐹 ↾ 𝐶) Fn dom (◡𝐹 ↾ 𝐶) ↔ (◡𝐹 ↾ 𝐶) Fn 𝐶))
10	3, 9	mpbid 231	. . . 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 6591	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun (𝐹 ↾ (◡𝐹 “ 𝐶)))
13		funcnvres2 6628	. . . . . . 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 6570	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (Fun ◡(◡𝐹 ↾ 𝐶) ↔ Fun (𝐹 ↾ (◡𝐹 “ 𝐶))))
16	12, 15	mpbird 256	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → Fun ◡(◡𝐹 ↾ 𝐶))
17		df-ima 5689	. . . . . 6 ⊢ (◡𝐹 “ 𝐶) = ran (◡𝐹 ↾ 𝐶)
18	17	eqcomi 2741	. . . . 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 6838	. . . 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 1343	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶))
22		f1ocnv 6845	. . 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 6821	. . 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 231	1 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1087   = wceq 1541   ⊆ wss 3948  ^◡ccnv 5675  dom cdm 5676  ran crn 5677   ↾ cres 5678   “ cima 5679  Fun wfun 6537   Fn wfn 6538  –1-1-onto→wf1o 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550

This theorem is referenced by:  carsggect  33312