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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ispsubclN | Structured version Visualization version GIF version |
Description: The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
psubclset.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
psubclset.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
ispsubclN | ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psubclset.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | psubclset.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
3 | psubclset.c | . . . 4 ⊢ 𝐶 = (PSubCl‘𝐾) | |
4 | 1, 2, 3 | psubclsetN 39919 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐶 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)}) |
5 | 4 | eleq2d 2825 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)})) |
6 | 1 | fvexi 6921 | . . . . 5 ⊢ 𝐴 ∈ V |
7 | 6 | ssex 5327 | . . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ V) |
9 | sseq1 4021 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴)) | |
10 | 2fveq3 6912 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋))) | |
11 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
12 | 10, 11 | eqeq12d 2751 | . . . 4 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
13 | 9, 12 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
14 | 8, 13 | elab3 3689 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)} ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) |
15 | 5, 14 | bitrdi 287 | 1 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 Vcvv 3478 ⊆ wss 3963 ‘cfv 6563 Atomscatm 39245 ⊥𝑃cpolN 39885 PSubClcpscN 39917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-psubclN 39918 |
This theorem is referenced by: psubcliN 39921 psubcli2N 39922 0psubclN 39926 1psubclN 39927 atpsubclN 39928 pmapsubclN 39929 ispsubcl2N 39930 osumclN 39950 pexmidN 39952 pexmidlem6N 39958 |
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