Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fresf1o Structured version   Visualization version   GIF version

Theorem fresf1o 29663
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 6031 . . . . . . . 8 (Fun (𝐹𝐶) ↔ (𝐹𝐶) Fn dom (𝐹𝐶))
21biimpi 206 . . . . . . 7 (Fun (𝐹𝐶) → (𝐹𝐶) Fn dom (𝐹𝐶))
323ad2ant3 1127 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) Fn dom (𝐹𝐶))
4 simp2 1129 . . . . . . . . 9 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → 𝐶 ⊆ ran 𝐹)
5 df-rn 5229 . . . . . . . . 9 ran 𝐹 = dom 𝐹
64, 5syl6sseq 3757 . . . . . . . 8 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → 𝐶 ⊆ dom 𝐹)
7 ssdmres 5530 . . . . . . . 8 (𝐶 ⊆ dom 𝐹 ↔ dom (𝐹𝐶) = 𝐶)
86, 7sylib 208 . . . . . . 7 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → dom (𝐹𝐶) = 𝐶)
98fneq2d 6095 . . . . . 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 1128 . . . . . . 7 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun 𝐹)
12 funres 6042 . . . . . . 7 (Fun 𝐹 → Fun (𝐹 ↾ (𝐹𝐶)))
1311, 12syl 17 . . . . . 6 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → Fun (𝐹 ↾ (𝐹𝐶)))
14 funcnvres2 6082 . . . . . . . 8 (Fun 𝐹(𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
1511, 14syl 17 . . . . . . 7 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹𝐶) = (𝐹 ↾ (𝐹𝐶)))
1615funeqd 6023 . . . . . 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 5231 . . . . . . 7 (𝐹𝐶) = ran (𝐹𝐶)
1918eqcomi 2733 . . . . . 6 ran (𝐹𝐶) = (𝐹𝐶)
2019a1i 11 . . . . 5 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ran (𝐹𝐶) = (𝐹𝐶))
2110, 17, 203jca 1379 . . . 4 ((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → ((𝐹𝐶) Fn 𝐶 ∧ Fun (𝐹𝐶) ∧ ran (𝐹𝐶) = (𝐹𝐶)))
22 dff1o2 6255 . . . 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 6262 . . 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 6240 . . 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 1072   = wceq 1596  wss 3680  ccnv 5217  dom cdm 5218  ran crn 5219  cres 5220  cima 5221  Fun wfun 5995   Fn wfn 5996  1-1-ontowf1o 6000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008
This theorem is referenced by:  carsggect  30610
  Copyright terms: Public domain W3C validator