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Theorem fresf1o 31591
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 6532 . . . . . . 7 (Fun (𝐹𝐶) ↔ (𝐹𝐶) Fn dom (𝐹𝐶))
21biimpi 215 . . . . . 6 (Fun (𝐹𝐶) → (𝐹𝐶) Fn dom (𝐹𝐶))
323ad2ant3 1136 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) Fn dom (𝐹𝐶))
4 simp2 1138 . . . . . . . 8 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → 𝐶 ⊆ ran 𝐹)
5 df-rn 5645 . . . . . . . 8 ran 𝐹 = dom 𝐹
64, 5sseqtrdi 3995 . . . . . . 7 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → 𝐶 ⊆ dom 𝐹)
7 ssdmres 5961 . . . . . . 7 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
86, 7sylib 217 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → dom (𝐹𝐶) = 𝐶)
98fneq2d 6597 . . . . 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 1137 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun 𝐹)
1211funresd 6545 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun (𝐹 ↾ (𝐹𝐶)))
13 funcnvres2 6582 . . . . . . 7 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
1411, 13syl 17 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
1514funeqd 6524 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (Fun (𝐹𝐶) ↔ Fun (𝐹 ↾ (𝐹𝐶))))
1612, 15mpbird 257 . . . 4 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun (𝐹𝐶))
17 df-ima 5647 . . . . . 6 (𝐹𝐶) = ran (𝐹𝐶)
1817eqcomi 2742 . . . . 5 ran (𝐹𝐶) = (𝐹𝐶)
1918a1i 11 . . . 4 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ran (𝐹𝐶) = (𝐹𝐶))
20 dff1o2 6790 . . . 4 ((𝐹𝐶):𝐶1-1-onto→(𝐹𝐶) ↔ ((𝐹𝐶) Fn 𝐶 ∧ Fun (𝐹𝐶) ∧ ran (𝐹𝐶) = (𝐹𝐶)))
2110, 16, 19, 20syl3anbrc 1344 . . 3 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
22 f1ocnv 6797 . . 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 6773 . . 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 1088   = wceq 1542  wss 3911  ccnv 5633  dom cdm 5634  ran crn 5635  cres 5636  cima 5637  Fun wfun 6491   Fn wfn 6492  1-1-ontowf1o 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  carsggect  32975
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