Theorem ixpv 48995

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 6659	. . 3 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶V)
2		vex 3440	. . . 4 ⊢ 𝑔 ∈ V
3		fneq1 6578	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴))
4	2, 3	elab 3630	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐴} ↔ 𝑔 Fn 𝐴)
5	2	elixpconst 8835	. . 3 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 V ↔ 𝑔:𝐴⟶V)
6	1, 4, 5	3bitr4ri 304	. 2 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 V ↔ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐴})
7	6	eqriv 2728	1 ⊢ X𝑥 ∈ 𝐴 V = {𝑓 ∣ 𝑓 Fn 𝐴}

Syntax hints:   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436   Fn wfn 6482  ⟶wf 6483  Xcixp 8827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fv 6495  df-ixp 8828

This theorem is referenced by: (None)