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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 6687 . . . . 5 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
21ineq1d 4219 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ I ) = (𝐹 ∩ I ))
3 inres 6015 . . . . . 6 ( I ∩ (𝐹𝐴)) = (( I ∩ 𝐹) ↾ 𝐴)
4 incom 4209 . . . . . . 7 ( I ∩ 𝐹) = (𝐹 ∩ I )
54reseq1i 5993 . . . . . 6 (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴)
63, 5eqtri 2765 . . . . 5 ( I ∩ (𝐹𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
7 incom 4209 . . . . 5 ((𝐹𝐴) ∩ I ) = ( I ∩ (𝐹𝐴))
8 inres 6015 . . . . 5 (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
96, 7, 83eqtr4i 2775 . . . 4 ((𝐹𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))
102, 9eqtr3di 2792 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)))
1110dmeqd 5916 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴)))
12 fnresi 6697 . . 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 2748 . . . 4 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
1716rabbiia 3440 . . 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 2781 1 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436  cin 3950   I cid 5577  dom cdm 5685  cres 5687   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  fnelfp  7195
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