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Theorem fresf1o 32122
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 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶)

Proof of Theorem fresf1o
StepHypRef Expression
1 funfn 6577 . . . . . . 7 (Fun (𝐹𝐶) ↔ (𝐹𝐶) Fn dom (𝐹𝐶))
21biimpi 215 . . . . . 6 (Fun (𝐹𝐶) → (𝐹𝐶) Fn dom (𝐹𝐶))
323ad2ant3 1133 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) Fn dom (𝐹𝐶))
4 simp2 1135 . . . . . . . 8 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → 𝐶 ⊆ ran 𝐹)
5 df-rn 5686 . . . . . . . 8 ran 𝐹 = dom 𝐹
64, 5sseqtrdi 4031 . . . . . . 7 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → 𝐶 ⊆ dom 𝐹)
7 ssdmres 6003 . . . . . . 7 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
86, 7sylib 217 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → dom (𝐹𝐶) = 𝐶)
98fneq2d 6642 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ((𝐹𝐶) Fn dom (𝐹𝐶) ↔ (𝐹𝐶) Fn 𝐶))
103, 9mpbid 231 . . . 4 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) Fn 𝐶)
11 simp1 1134 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun 𝐹)
1211funresd 6590 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun (𝐹 ↾ (𝐹𝐶)))
13 funcnvres2 6627 . . . . . . 7 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
1411, 13syl 17 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
1514funeqd 6569 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (Fun (𝐹𝐶) ↔ Fun (𝐹 ↾ (𝐹𝐶))))
1612, 15mpbird 256 . . . 4 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun (𝐹𝐶))
17 df-ima 5688 . . . . . 6 (𝐹𝐶) = ran (𝐹𝐶)
1817eqcomi 2739 . . . . 5 ran (𝐹𝐶) = (𝐹𝐶)
1918a1i 11 . . . 4 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ran (𝐹𝐶) = (𝐹𝐶))
20 dff1o2 6837 . . . 4 ((𝐹𝐶):𝐶1-1-onto→(𝐹𝐶) ↔ ((𝐹𝐶) Fn 𝐶 ∧ Fun (𝐹𝐶) ∧ ran (𝐹𝐶) = (𝐹𝐶)))
2110, 16, 19, 20syl3anbrc 1341 . . 3 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
22 f1ocnv 6844 . . 3 ((𝐹𝐶):𝐶1-1-onto→(𝐹𝐶) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
2321, 22syl 17 . 2 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
24 f1oeq1 6820 . . 3 ((𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)) → ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 ↔ (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶))
2511, 13, 243syl 18 . 2 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 ↔ (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶))
2623, 25mpbid 231 1 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1085   = wceq 1539  wss 3947  ccnv 5674  dom cdm 5675  ran crn 5676  cres 5677  cima 5678  Fun wfun 6536   Fn wfn 6537  1-1-ontowf1o 6541
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549
This theorem is referenced by:  carsggect  33615
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