Theorem elixpconstg 45125

Description: Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Assertion

Ref	Expression
elixpconstg	⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elixpconstg

Step	Hyp	Ref	Expression
1		ixpfn 8827	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴)
2		elixp2 8825	. . . 4 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
3	2	simp3bi 1147	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
4		ffnfv 7052	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
5	1, 3, 4	sylanbrc 583	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶𝐵)
6		elex 3457	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
7	6	adantr 480	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ V)
8		ffn 6651	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
9	8	adantl 481	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 Fn 𝐴)
10	4	simprbi 496	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
11	10	adantl 481	. . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
12	7, 9, 11, 2	syl3anbrc 1344	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵)
13	12	ex 412	. 2 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴⟶𝐵 → 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵))
14	5, 13	impbid2 226	1 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  Xcixp 8821

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 5234  ax-nul 5244  ax-pr 5370

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 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ixp 8822

This theorem is referenced by:  iinhoiicclem  46710