Theorem ixpv 50208

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 6733	. . 3 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶V)
2		vex 3458	. . . 4 ⊢ 𝑔 ∈ V
3		fneq1 6652	. . . 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 8961	. . 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 2760	1 ⊢ X𝑥 ∈ 𝐴 V = {𝑓 ∣ 𝑓 Fn 𝐴}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {cab 2741  Vcvv 3454   Fn wfn 6553  ⟶wf 6554  Xcixp 8953

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-ixp 8954

This theorem is used by: (None)