Theorem ixpm 6789

Description: If an infinite Cartesian product of a family 𝐵(𝑥) is inhabited, every 𝐵(𝑥) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)

Assertion

Ref	Expression
ixpm	⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)

Distinct variable groups:   𝐴,𝑓   𝑧,𝑓,𝐵   𝑥,𝑓,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐵(𝑥)

Proof of Theorem ixpm

Step	Hyp	Ref	Expression
1		df-ixp 6758	. . . 4 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
2	1	abeq2i 2307	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
3		elex2 2779	. . . 4 ⊢ ((𝑓‘𝑥) ∈ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵)
4	3	ralimi 2560	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
5	2, 4	simplbiim 387	. 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
6	5	exlimiv 1612	1 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1506   ∈ wcel 2167  {cab 2182  ∀wral 2475   Fn wfn 5253  ‘cfv 5258  Xcixp 6757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-v 2765  df-ixp 6758

This theorem is referenced by:  ixp0  6790