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Theorem fninfp 6928
Description: Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fninfp (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥})
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem fninfp
StepHypRef Expression
1 inres 5864 . . . . . 6 ( I ∩ (𝐹𝐴)) = (( I ∩ 𝐹) ↾ 𝐴)
2 incom 4175 . . . . . . 7 ( I ∩ 𝐹) = (𝐹 ∩ I )
32reseq1i 5842 . . . . . 6 (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴)
41, 3eqtri 2841 . . . . 5 ( I ∩ (𝐹𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
5 incom 4175 . . . . 5 ((𝐹𝐴) ∩ I ) = ( I ∩ (𝐹𝐴))
6 inres 5864 . . . . 5 (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
74, 5, 63eqtr4i 2851 . . . 4 ((𝐹𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))
8 fnresdm 6459 . . . . 5 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
98ineq1d 4185 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ I ) = (𝐹 ∩ I ))
107, 9syl5reqr 2868 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)))
1110dmeqd 5767 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴)))
12 fnresi 6469 . . 3 ( I ↾ 𝐴) Fn 𝐴
13 fndmin 6807 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)})
1412, 13mpan2 687 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)})
15 fvresi 6927 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
1615eqeq2d 2829 . . . 4 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
1716rabbiia 3470 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)} = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥}
1817a1i 11 . 2 (𝐹 Fn 𝐴 → {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)} = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥})
1911, 14, 183eqtrd 2857 1 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {crab 3139  cin 3932   I cid 5452  dom cdm 5548  cres 5550   Fn wfn 6343  cfv 6348
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-sbc 3770  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-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  fnelfp  6929
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