Theorem fnelfp 7180

Description: Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
fnelfp	⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑋) = 𝑋))

Proof of Theorem fnelfp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fninfp 7179	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥})
2	1	eleq2d 2851	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ 𝑋 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}))
3		fveq2 6886	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
4		id 23	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
5	3, 4	eqeq12d 2781	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑋) = 𝑋))
6	5	elrab3 3653	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥} ↔ (𝐹‘𝑋) = 𝑋))
7	2, 6	sylan9bb 519	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑋) = 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3418   ∩ cin 3905   I cid 5557  dom cdm 5663   Fn wfn 6536  ‘cfv 6541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-res 5675  df-iota 6497  df-fun 6543  df-fn 6544  df-fv 6549

This theorem is used by:  nfpconfp  33053  ismrcd1  43507  ismrcd2  43508  istopclsd  43509