Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dffix2 Structured version   Visualization version   GIF version

Theorem dffix2 32601
Description: The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
dffix2 Fix 𝐴 = ran (𝐴 ∩ I )

Proof of Theorem dffix2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3401 . . . 4 𝑥 ∈ V
21elfix 32599 . . 3 (𝑥 Fix 𝐴𝑥𝐴𝑥)
31elrn 5612 . . . 4 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥)
4 brin 4938 . . . . . 6 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥𝑦 I 𝑥))
5 ancom 454 . . . . . 6 ((𝑦𝐴𝑥𝑦 I 𝑥) ↔ (𝑦 I 𝑥𝑦𝐴𝑥))
61ideq 5520 . . . . . . 7 (𝑦 I 𝑥𝑦 = 𝑥)
76anbi1i 617 . . . . . 6 ((𝑦 I 𝑥𝑦𝐴𝑥) ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
84, 5, 73bitri 289 . . . . 5 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
98exbii 1892 . . . 4 (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥))
10 breq1 4889 . . . . 5 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝑥𝐴𝑥))
111, 10ceqsexv 3444 . . . 4 (∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥)
123, 9, 113bitri 289 . . 3 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥)
132, 12bitr4i 270 . 2 (𝑥 Fix 𝐴𝑥 ∈ ran (𝐴 ∩ I ))
1413eqriv 2775 1 Fix 𝐴 = ran (𝐴 ∩ I )
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1601  wex 1823  wcel 2107  cin 3791   class class class wbr 4886   I cid 5260  ran crn 5356   Fix cfix 32531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-dm 5365  df-rn 5366  df-fix 32555
This theorem is referenced by:  fixssrn  32603
  Copyright terms: Public domain W3C validator