Theorem fxpval 33120

Description: Value of the set of fixed points. (Contributed by Thierry Arnoux, 18-Nov-2025.)

Hypotheses

Ref	Expression
fxpval.1	⊢ (𝜑 → 𝐵 ∈ 𝑉)
fxpval.2	⊢ (𝜑 → 𝐴 ∈ 𝑊)

Assertion

Ref	Expression
fxpval	⊢ (𝜑 → (𝐵FixPts𝐴) = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})

Distinct variable groups:   𝐴,𝑝,𝑥   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑝)   𝐵(𝑝)   𝑉(𝑥,𝑝)   𝑊(𝑥,𝑝)

Proof of Theorem fxpval

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fxp 33119	. . 3 ⊢ FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥})
2	1	a1i 11	. 2 ⊢ (𝜑 → FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥}))
3		simpl 482	. . . 4 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐴) → 𝑏 = 𝐵)
4		dmeq 5850	. . . . . . 7 ⊢ (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
5	4	dmeqd 5852	. . . . . 6 ⊢ (𝑎 = 𝐴 → dom dom 𝑎 = dom dom 𝐴)
6		oveq 7359	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑝𝑎𝑥) = (𝑝𝐴𝑥))
7	6	eqeq1d 2731	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑝𝑎𝑥) = 𝑥 ↔ (𝑝𝐴𝑥) = 𝑥))
8	5, 7	raleqbidv 3310	. . . . 5 ⊢ (𝑎 = 𝐴 → (∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥 ↔ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥))
9	8	adantl 481	. . . 4 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐴) → (∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥 ↔ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥))
10	3, 9	rabeqbidv 3415	. . 3 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐴) → {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥} = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
11	10	adantl 481	. 2 ⊢ ((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) → {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥} = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
12		fxpval.1	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
13	12	elexd 3462	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
14		fxpval.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
15	14	elexd 3462	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
16		eqid 2729	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥}
17	16, 12	rabexd 5282	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} ∈ V)
18	2, 11, 13, 15, 17	ovmpod 7505	1 ⊢ (𝜑 → (𝐵FixPts𝐴) = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396  Vcvv 3438  dom cdm 5623  (class class class)co 7353   ∈ cmpo 7355  FixPtscfxp 33118

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 5238  ax-nul 5248  ax-pr 5374

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-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-fxp 33119

This theorem is referenced by:  fxpss  33121  fxpgaval  33122