Theorem fxpval 33605

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 33604	. . 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 488	. . . 4 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐴) → 𝑏 = 𝐵)
4		dmeq 5887	. . . . . . 7 ⊢ (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
5	4	dmeqd 5889	. . . . . 6 ⊢ (𝑎 = 𝐴 → dom dom 𝑎 = dom dom 𝐴)
6		oveq 7419	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑝𝑎𝑥) = (𝑝𝐴𝑥))
7	6	eqeq1d 2762	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑝𝑎𝑥) = 𝑥 ↔ (𝑝𝐴𝑥) = 𝑥))
8	5, 7	raleqbidv 3334	. . . . 5 ⊢ (𝑎 = 𝐴 → (∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥 ↔ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥))
9	8	adantl 487	. . . 4 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐴) → (∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥 ↔ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥))
10	3, 9	rabeqbidv 3429	. . 3 ⊢ ((𝑏 = 𝐵 ∧ 𝑎 = 𝐴) → {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥} = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
11	10	adantl 487	. 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 3473	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
14		fxpval.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
15	14	elexd 3473	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
16		eqid 2760	. . 3 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥}
17	16, 12	rabexd 5304	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} ∈ V)
18	2, 11, 13, 15, 17	ovmpod 7565	1 ⊢ (𝜑 → (𝐵FixPts𝐴) = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450  dom cdm 5655  (class class class)co 7413   ∈ cmpo 7415  FixPtscfxp 33603

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-fxp 33604

This theorem is used by:  fxpss  33606  fxpgaval  33607