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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 4811 | . . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴) | |
2 | isptfin.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
3 | 1, 2 | eqtr4di 2851 | . . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋) |
4 | rabeq 3431 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦}) | |
5 | 4 | eleq1d 2874 | . . 3 ⊢ (𝑎 = 𝐴 → ({𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
6 | 3, 5 | raleqbidv 3354 | . 2 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
7 | df-ptfin 22111 | . 2 ⊢ PtFin = {𝑎 ∣ ∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin} | |
8 | 6, 7 | elab2g 3616 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ∪ cuni 4800 Fincfn 8492 PtFincptfin 22108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 df-ptfin 22111 |
This theorem is referenced by: finptfin 22123 ptfinfin 22124 lfinpfin 22129 |
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