Theorem ixpm 6578

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 6547	. . . 4 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
2	1	abeq2i 2225	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
3		elex2 2673	. . . 4 ⊢ ((𝑓‘𝑥) ∈ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵)
4	3	ralimi 2469	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
5	2, 4	simplbiim 382	. 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)
6	5	exlimiv 1560	1 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1451   ∈ wcel 1463  {cab 2101  ∀wral 2390   Fn wfn 5076  ‘cfv 5081  Xcixp 6546

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2395  df-v 2659  df-ixp 6547

This theorem is referenced by:  ixp0  6579