![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4714 | . . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴) | |
2 | isptfin.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐴 | |
3 | 1, 2 | syl6eqr 2826 | . . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋) |
4 | rabeq 3400 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦}) | |
5 | 4 | eleq1d 2844 | . . 3 ⊢ (𝑎 = 𝐴 → ({𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
6 | 3, 5 | raleqbidv 3335 | . 2 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
7 | df-ptfin 21808 | . 2 ⊢ PtFin = {𝑎 ∣ ∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin} | |
8 | 6, 7 | elab2g 3578 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2048 ∀wral 3082 {crab 3086 ∪ cuni 4706 Fincfn 8298 PtFincptfin 21805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-uni 4707 df-ptfin 21808 |
This theorem is referenced by: finptfin 21820 ptfinfin 21821 lfinpfin 21826 |
Copyright terms: Public domain | W3C validator |