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Theorem fninfp 6481
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 5449 . . . . . 6 ( I ∩ (𝐹𝐴)) = (( I ∩ 𝐹) ↾ 𝐴)
2 incom 3838 . . . . . . 7 ( I ∩ 𝐹) = (𝐹 ∩ I )
32reseq1i 5424 . . . . . 6 (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴)
41, 3eqtri 2673 . . . . 5 ( I ∩ (𝐹𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
5 incom 3838 . . . . 5 ((𝐹𝐴) ∩ I ) = ( I ∩ (𝐹𝐴))
6 inres 5449 . . . . 5 (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
74, 5, 63eqtr4i 2683 . . . 4 ((𝐹𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))
8 fnresdm 6038 . . . . 5 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
98ineq1d 3846 . . . 4 (𝐹 Fn 𝐴 → ((𝐹𝐴) ∩ I ) = (𝐹 ∩ I ))
107, 9syl5reqr 2700 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)))
1110dmeqd 5358 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴)))
12 fnresi 6046 . . 3 ( I ↾ 𝐴) Fn 𝐴
13 fndmin 6364 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)})
1412, 13mpan2 707 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)})
15 fvresi 6480 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
1615eqeq2d 2661 . . . 4 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
1716rabbiia 3215 . . 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 2689 1 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {crab 2945  cin 3606   I cid 5052  dom cdm 5143  cres 5145   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  fnelfp  6482
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