MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resin Structured version   Visualization version   GIF version

Theorem resin 6320
Description: The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
Assertion
Ref Expression
resin ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))

Proof of Theorem resin
StepHypRef Expression
1 resdif 6319 . . . 4 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
2 f1ofo 6306 . . . 4 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐶𝐷))
31, 2syl 17 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐶𝐷))
4 resdif 6319 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐶𝐷)) → (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
53, 4syld3an3 1516 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
6 dfin4 4010 . . . 4 (𝐶𝐷) = (𝐶 ∖ (𝐶𝐷))
7 f1oeq3 6291 . . . 4 ((𝐶𝐷) = (𝐶 ∖ (𝐶𝐷)) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶 ∖ (𝐶𝐷))))
86, 7ax-mp 5 . . 3 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
9 dfin4 4010 . . . 4 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
10 f1oeq2 6290 . . . 4 ((𝐴𝐵) = (𝐴 ∖ (𝐴𝐵)) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷))))
119, 10ax-mp 5 . . 3 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
129reseq2i 5548 . . . 4 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴𝐵)))
13 f1oeq1 6289 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷))))
1412, 13ax-mp 5 . . 3 ((𝐹 ↾ (𝐴𝐵)):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
158, 11, 143bitrri 287 . 2 ((𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
165, 15sylib 208 1 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1632  cdif 3712  cin 3714  ccnv 5265  cres 5268  Fun wfun 6043  ontowfo 6047  1-1-ontowf1o 6048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator