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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpmapat | Structured version Visualization version GIF version |
Description: Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.) |
Ref | Expression |
---|---|
pmapat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pmapat.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
elpmapat | ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑋 ∈ (𝑀‘𝑃) ↔ 𝑋 = 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | pmapat.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
3 | 1, 2 | pmapat 36901 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑀‘𝑃) = {𝑃}) |
4 | 3 | eleq2d 2900 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑋 ∈ (𝑀‘𝑃) ↔ 𝑋 ∈ {𝑃})) |
5 | elsn2g 4605 | . . 3 ⊢ (𝑃 ∈ 𝐴 → (𝑋 ∈ {𝑃} ↔ 𝑋 = 𝑃)) | |
6 | 5 | adantl 484 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑋 ∈ {𝑃} ↔ 𝑋 = 𝑃)) |
7 | 4, 6 | bitrd 281 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑋 ∈ (𝑀‘𝑃) ↔ 𝑋 = 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4569 ‘cfv 6357 Atomscatm 36401 HLchlt 36488 pmapcpmap 36635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-proset 17540 df-poset 17558 df-plt 17570 df-glb 17587 df-p0 17651 df-lat 17658 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-pmap 36642 |
This theorem is referenced by: pmapjat1 36991 |
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