Theorem ixpv 49803

Description: Infinite Cartesian product of the universal class is the set of functions with a fixed domain. (Contributed by Zhi Wang, 1-Nov-2025.)

Assertion

Ref	Expression
ixpv	⊢ X𝑥 ∈ 𝐴 V = {𝑓 ∣ 𝑓 Fn 𝐴}

Distinct variable group:   𝐴,𝑓,𝑥

Proof of Theorem ixpv

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn2 6708	. . 3 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶V)
2		vex 3457	. . . 4 ⊢ 𝑔 ∈ V
3		fneq1 6627	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴))
4	2, 3	elab 3636	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐴} ↔ 𝑔 Fn 𝐴)
5	2	elixpconst 8915	. . 3 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 V ↔ 𝑔:𝐴⟶V)
6	1, 4, 5	3bitr4ri 307	. 2 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 V ↔ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐴})
7	6	eqriv 2759	1 ⊢ X𝑥 ∈ 𝐴 V = {𝑓 ∣ 𝑓 Fn 𝐴}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  {cab 2740  Vcvv 3453   Fn wfn 6532  ⟶wf 6533  Xcixp 8907

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ixp 8908

This theorem is used by: (None)