Theorem fninfp 7151

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 6640	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
2	1	ineq1d 4185	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ I ))
3		inres 5971	. . . . . 6 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = (( I ∩ 𝐹) ↾ 𝐴)
4		incom 4175	. . . . . . 7 ⊢ ( I ∩ 𝐹) = (𝐹 ∩ I )
5	4	reseq1i 5949	. . . . . 6 ⊢ (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴)
6	3, 5	eqtri 2753	. . . . 5 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
7		incom 4175	. . . . 5 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = ( I ∩ (𝐹 ↾ 𝐴))
8		inres 5971	. . . . 5 ⊢ (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴)
9	6, 7, 8	3eqtr4i 2763	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))
10	2, 9	eqtr3di 2780	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)))
11	10	dmeqd 5872	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴)))
12		fnresi 6650	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
13		fndmin 7020	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)})
14	12, 13	mpan2 691	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)})
15		fvresi 7150	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
16	15	eqeq2d 2741	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) = 𝑥))
17	16	rabbiia 3412	. . 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 2769	1 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3408   ∩ cin 3916   I cid 5535  dom cdm 5641   ↾ cres 5643   Fn wfn 6509  ‘cfv 6514

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by:  fnelfp  7152