Theorem fnelfp 7196

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 7195	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥})
2	1	eleq2d 2826	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ 𝑋 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}))
3		fveq2 6905	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
4		id 22	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
5	3, 4	eqeq12d 2752	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑋) = 𝑋))
6	5	elrab3 3692	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥} ↔ (𝐹‘𝑋) = 𝑋))
7	2, 6	sylan9bb 509	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑋) = 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {crab 3435   ∩ cin 3949   I cid 5576  dom cdm 5684   Fn wfn 6555  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568

This theorem is referenced by:  nfpconfp  32643  ismrcd1  42714  ismrcd2  42715  istopclsd  42716