Theorem fnelfp 7149

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 7148	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥})
2	1	eleq2d 2814	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ 𝑋 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}))
3		fveq2 6858	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
4		id 22	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
5	3, 4	eqeq12d 2745	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑋) = 𝑋))
6	5	elrab3 3660	. 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 1540   ∈ wcel 2109  {crab 3405   ∩ cin 3913   I cid 5532  dom cdm 5638   Fn wfn 6506  ‘cfv 6511

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by:  nfpconfp  32556  ismrcd1  42686  ismrcd2  42687  istopclsd  42688