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Theorem fresf1o 29277
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 5877 . . . . . . . 8 (Fun (𝐹𝐶) ↔ (𝐹𝐶) Fn dom (𝐹𝐶))
21biimpi 206 . . . . . . 7 (Fun (𝐹𝐶) → (𝐹𝐶) Fn dom (𝐹𝐶))
323ad2ant3 1082 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) Fn dom (𝐹𝐶))
4 simp2 1060 . . . . . . . . 9 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → 𝐶 ⊆ ran 𝐹)
5 df-rn 5085 . . . . . . . . 9 ran 𝐹 = dom 𝐹
64, 5syl6sseq 3630 . . . . . . . 8 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → 𝐶 ⊆ dom 𝐹)
7 ssdmres 5379 . . . . . . . 8 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
86, 7sylib 208 . . . . . . 7 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → dom (𝐹𝐶) = 𝐶)
98fneq2d 5940 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ((𝐹𝐶) Fn dom (𝐹𝐶) ↔ (𝐹𝐶) Fn 𝐶))
103, 9mpbid 222 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) Fn 𝐶)
11 simp1 1059 . . . . . . 7 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun 𝐹)
12 funres 5887 . . . . . . 7 (Fun 𝐹 → Fun (𝐹 ↾ (𝐹𝐶)))
1311, 12syl 17 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun (𝐹 ↾ (𝐹𝐶)))
14 funcnvres2 5927 . . . . . . . 8 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
1511, 14syl 17 . . . . . . 7 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
1615funeqd 5869 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (Fun (𝐹𝐶) ↔ Fun (𝐹 ↾ (𝐹𝐶))))
1713, 16mpbird 247 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun (𝐹𝐶))
18 df-ima 5087 . . . . . . 7 (𝐹𝐶) = ran (𝐹𝐶)
1918eqcomi 2630 . . . . . 6 ran (𝐹𝐶) = (𝐹𝐶)
2019a1i 11 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ran (𝐹𝐶) = (𝐹𝐶))
2110, 17, 203jca 1240 . . . 4 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ((𝐹𝐶) Fn 𝐶 ∧ Fun (𝐹𝐶) ∧ ran (𝐹𝐶) = (𝐹𝐶)))
22 dff1o2 6099 . . . 4 ((𝐹𝐶):𝐶1-1-onto→(𝐹𝐶) ↔ ((𝐹𝐶) Fn 𝐶 ∧ Fun (𝐹𝐶) ∧ ran (𝐹𝐶) = (𝐹𝐶)))
2321, 22sylibr 224 . . 3 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
24 f1ocnv 6106 . . 3 ((𝐹𝐶):𝐶1-1-onto→(𝐹𝐶) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
2523, 24syl 17 . 2 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶)
26 f1oeq1 6084 . . 3 ((𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)) → ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 ↔ (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶))
2711, 14, 263syl 18 . 2 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ((𝐹𝐶):(𝐹𝐶)–1-1-onto𝐶 ↔ (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶))
2825, 27mpbid 222 1 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wss 3555  ccnv 5073  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  Fun wfun 5841   Fn wfn 5842  1-1-ontowf1o 5846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854
This theorem is referenced by:  carsggect  30161
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