Theorem ixpv 49502

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 6693	. . 3 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶V)
2		vex 3459	. . . 4 ⊢ 𝑔 ∈ V
3		fneq1 6612	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴))
4	2, 3	elab 3639	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐴} ↔ 𝑔 Fn 𝐴)
5	2	elixpconst 8887	. . 3 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 V ↔ 𝑔:𝐴⟶V)
6	1, 4, 5	3bitr4ri 306	. 2 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 V ↔ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐴})
7	6	eqriv 2760	1 ⊢ X𝑥 ∈ 𝐴 V = {𝑓 ∣ 𝑓 Fn 𝐴}

Syntax hints:   = wceq 1561   ∈ wcel 2143  {cab 2741  Vcvv 3455   Fn wfn 6516  ⟶wf 6517  Xcixp 8879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ixp 8880

This theorem is referenced by: (None)