![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > elpmap | Structured version Visualization version GIF version |
Description: Member of a projective map. (Contributed by NM, 27-Jan-2012.) |
Ref | Expression |
---|---|
pmapfval.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapfval.l | ⊢ ≤ = (le‘𝐾) |
pmapfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pmapfval.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
elpmap | ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ (𝑀‘𝑋) ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | pmapfval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | pmapfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | pmapfval.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
5 | 1, 2, 3, 4 | pmapval 37053 | . . 3 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑥 ∈ 𝐴 ∣ 𝑥 ≤ 𝑋}) |
6 | 5 | eleq2d 2875 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ (𝑀‘𝑋) ↔ 𝑃 ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ≤ 𝑋})) |
7 | breq1 5033 | . . 3 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋)) | |
8 | 7 | elrab 3628 | . 2 ⊢ (𝑃 ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ≤ 𝑋} ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑋)) |
9 | 6, 8 | syl6bb 290 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ (𝑀‘𝑋) ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 lecple 16564 Atomscatm 36559 pmapcpmap 36793 |
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 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-pmap 36800 |
This theorem is referenced by: pmapjoin 37148 pmapjat1 37149 |
Copyright terms: Public domain | W3C validator |