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Theorem dffix2 33263
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 3495 . . . 4 𝑥 ∈ V
21elfix 33261 . . 3 (𝑥 Fix 𝐴𝑥𝐴𝑥)
31elrn 5815 . . . 4 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥)
4 brin 5109 . . . . . 6 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥𝑦 I 𝑥))
5 ancom 461 . . . . . 6 ((𝑦𝐴𝑥𝑦 I 𝑥) ↔ (𝑦 I 𝑥𝑦𝐴𝑥))
61ideq 5716 . . . . . . 7 (𝑦 I 𝑥𝑦 = 𝑥)
76anbi1i 623 . . . . . 6 ((𝑦 I 𝑥𝑦𝐴𝑥) ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
84, 5, 73bitri 298 . . . . 5 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
98exbii 1839 . . . 4 (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥))
10 breq1 5060 . . . . 5 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝑥𝐴𝑥))
1110equsexvw 2002 . . . 4 (∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥)
123, 9, 113bitri 298 . . 3 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥)
132, 12bitr4i 279 . 2 (𝑥 Fix 𝐴𝑥 ∈ ran (𝐴 ∩ I ))
1413eqriv 2815 1 Fix 𝐴 = ran (𝐴 ∩ I )
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wex 1771  wcel 2105  cin 3932   class class class wbr 5057   I cid 5452  ran crn 5549   Fix cfix 33193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-fix 33217
This theorem is referenced by:  fixssrn  33265
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