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Theorem fninfp 6323
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 5321 . . . . . 6 ( I ∩ (𝐹𝐴)) = (( I ∩ 𝐹) ↾ 𝐴)
2 incom 3767 . . . . . . 7 ( I ∩ 𝐹) = (𝐹 ∩ I )
32reseq1i 5300 . . . . . 6 (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴)
41, 3eqtri 2632 . . . . 5 ( I ∩ (𝐹𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
5 incom 3767 . . . . 5 ((𝐹𝐴) ∩ I ) = ( I ∩ (𝐹𝐴))
6 inres 5321 . . . . 5 (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
74, 5, 63eqtr4i 2642 . . . 4 ((𝐹𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))
8 fnresdm 5900 . . . . 5 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
98ineq1d 3775 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ I ) = (𝐹 ∩ I ))
107, 9syl5reqr 2659 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)))
1110dmeqd 5235 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴)))
12 fnresi 5908 . . 3 ( I ↾ 𝐴) Fn 𝐴
13 fndmin 6217 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)})
1412, 13mpan2 703 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)})
15 fvresi 6322 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
1615eqeq2d 2620 . . . 4 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
1716rabbiia 3161 . . 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 2648 1 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  {crab 2900  cin 3539   I cid 4938  dom cdm 5028  cres 5030   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-res 5040  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  fnelfp  6324
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