Theorem fninfp 7169

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 ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥})

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem fninfp

Step	Hyp	Ref	Expression
1		fnresdm 6667	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	ineq1d 4211	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ I ))
3		inres 5998	. . . . . 6 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = (( I ∩ 𝐹) ↾ 𝐴)
4		incom 4201	. . . . . . 7 ⊢ ( I ∩ 𝐹) = (𝐹 ∩ I )
5	4	reseq1i 5976	. . . . . 6 ⊢ (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴)
6	3, 5	eqtri 2761	. . . . 5 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
7		incom 4201	. . . . 5 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = ( I ∩ (𝐹 ↾ 𝐴))
8		inres 5998	. . . . 5 ⊢ (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
9	6, 7, 8	3eqtr4i 2771	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))
10	2, 9	eqtr3di 2788	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)))
11	10	dmeqd 5904	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴)))
12		fnresi 6677	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
13		fndmin 7044	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)})
14	12, 13	mpan2 690	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)})
15		fvresi 7168	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
16	15	eqeq2d 2744	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) = 𝑥))
17	16	rabbiia 3437	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}
18	17	a1i 11	. 2 ⊢ (𝐹 Fn 𝐴 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥})
19	11, 14, 18	3eqtrd 2777	1 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  {crab 3433   ∩ cin 3947   I cid 5573  dom cdm 5676   ↾ cres 5678   Fn wfn 6536  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6493  df-fun 6543  df-fn 6544  df-fv 6549

This theorem is referenced by:  fnelfp  7170