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Theorem fninfp 7194
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 fnresdm 6688 . . . . 5 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
21ineq1d 4227 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ I ) = (𝐹 ∩ I ))
3 inres 6018 . . . . . 6 ( I ∩ (𝐹𝐴)) = (( I ∩ 𝐹) ↾ 𝐴)
4 incom 4217 . . . . . . 7 ( I ∩ 𝐹) = (𝐹 ∩ I )
54reseq1i 5996 . . . . . 6 (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴)
63, 5eqtri 2763 . . . . 5 ( I ∩ (𝐹𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
7 incom 4217 . . . . 5 ((𝐹𝐴) ∩ I ) = ( I ∩ (𝐹𝐴))
8 inres 6018 . . . . 5 (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
96, 7, 83eqtr4i 2773 . . . 4 ((𝐹𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))
102, 9eqtr3di 2790 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)))
1110dmeqd 5919 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴)))
12 fnresi 6698 . . 3 ( I ↾ 𝐴) Fn 𝐴
13 fndmin 7065 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)})
1412, 13mpan2 691 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)})
15 fvresi 7193 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
1615eqeq2d 2746 . . . 4 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
1716rabbiia 3437 . . 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 2779 1 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  cin 3962   I cid 5582  dom cdm 5689  cres 5691   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  fnelfp  7195
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