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Mirrors > Home > ILE Home > Th. List > ixpm | GIF version |
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.) |
Ref | Expression |
---|---|
ixpm | ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ixp 6656 | . . . 4 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
2 | 1 | abeq2i 2275 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
3 | elex2 2737 | . . . 4 ⊢ ((𝑓‘𝑥) ∈ 𝐵 → ∃𝑧 𝑧 ∈ 𝐵) | |
4 | 3 | ralimi 2527 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵) |
5 | 2, 4 | simplbiim 385 | . 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵) |
6 | 5 | exlimiv 1585 | 1 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1479 ∈ wcel 2135 {cab 2150 ∀wral 2442 Fn wfn 5177 ‘cfv 5182 Xcixp 6655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-ral 2447 df-v 2723 df-ixp 6656 |
This theorem is referenced by: ixp0 6688 |
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