Theorem ixpv 48759

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 6705	. . 3 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶V)
2		vex 3461	. . . 4 ⊢ 𝑔 ∈ V
3		fneq1 6626	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐴 ↔ 𝑔 Fn 𝐴))
4	2, 3	elab 3656	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐴} ↔ 𝑔 Fn 𝐴)
5	2	elixpconst 8914	. . 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 2731	1 ⊢ X𝑥 ∈ 𝐴 V = {𝑓 ∣ 𝑓 Fn 𝐴}

Syntax hints:   = wceq 1539   ∈ wcel 2107  {cab 2712  Vcvv 3457   Fn wfn 6523  ⟶wf 6524  Xcixp 8906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pr 5400

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-fv 6536  df-ixp 8907

This theorem is referenced by: (None)