| 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 39938 | . . 3 ⊢ (𝐾 ∈ 𝐷 → 𝐶 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)}) |
| 5 | 4 | eleq2d 2827 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)})) |
| 6 | 1 | fvexi 6920 | . . . . 5 ⊢ 𝐴 ∈ V |
| 7 | 6 | ssex 5321 | . . . 4 ⊢ (𝑋 ⊆ 𝐴 → 𝑋 ∈ V) |
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → 𝑋 ∈ V) |
| 9 | sseq1 4009 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴)) | |
| 10 | 2fveq3 6911 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋))) | |
| 11 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
| 12 | 10, 11 | eqeq12d 2753 | . . . 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 3686 | . 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 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 ⊆ wss 3951 ‘cfv 6561 Atomscatm 39264 ⊥𝑃cpolN 39904 PSubClcpscN 39936 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-psubclN 39937 |
| This theorem is referenced by: psubcliN 39940 psubcli2N 39941 0psubclN 39945 1psubclN 39946 atpsubclN 39947 pmapsubclN 39948 ispsubcl2N 39949 osumclN 39969 pexmidN 39971 pexmidlem6N 39977 |
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