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Mirrors > Home > MPE Home > Th. List > isptfin | Structured version Visualization version GIF version |
Description: The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.) |
Ref | Expression |
---|---|
isptfin.1 | ⊢ 𝑋 = ∪ 𝐴 |
Ref | Expression |
---|---|
isptfin | ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4380 | . . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴) | |
2 | isptfin.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
3 | 1, 2 | syl6eqr 2662 | . . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋) |
4 | rabeq 3166 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦}) | |
5 | 4 | eleq1d 2672 | . . 3 ⊢ (𝑎 = 𝐴 → ({𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
6 | 3, 5 | raleqbidv 3129 | . 2 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
7 | df-ptfin 21119 | . 2 ⊢ PtFin = {𝑎 ∣ ∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin} | |
8 | 6, 7 | elab2g 3322 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∪ cuni 4372 Fincfn 7841 PtFincptfin 21116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-uni 4373 df-ptfin 21119 |
This theorem is referenced by: finptfin 21131 ptfinfin 21132 lfinpfin 21137 |
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