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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 4913 | . . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴) | |
2 | isptfin.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
3 | 1, 2 | eqtr4di 2784 | . . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋) |
4 | rabeq 3440 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦}) | |
5 | 4 | eleq1d 2812 | . . 3 ⊢ (𝑎 = 𝐴 → ({𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
6 | 3, 5 | raleqbidv 3336 | . 2 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
7 | df-ptfin 23361 | . 2 ⊢ PtFin = {𝑎 ∣ ∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin} | |
8 | 6, 7 | elab2g 3665 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∀wral 3055 {crab 3426 ∪ cuni 4902 Fincfn 8938 PtFincptfin 23358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rab 3427 df-v 3470 df-in 3950 df-ss 3960 df-uni 4903 df-ptfin 23361 |
This theorem is referenced by: finptfin 23373 ptfinfin 23374 lfinpfin 23379 |
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