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Mirrors > Home > MPE Home > Th. List > Mathboxes > dicelval3 | Structured version Visualization version GIF version |
Description: Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.) |
Ref | Expression |
---|---|
dicval.l | ⊢ ≤ = (le‘𝐾) |
dicval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dicval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dicval.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
dicval.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dicval.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dicval.i | ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) |
dicval2.g | ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄) |
Ref | Expression |
---|---|
dicelval3 | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ ∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dicval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | dicval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | dicval.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dicval.p | . . . 4 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
5 | dicval.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | dicval.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
7 | dicval.i | . . . 4 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊) | |
8 | dicval2.g | . . . 4 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | dicval2 39193 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}) |
10 | 9 | eleq2d 2824 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})) |
11 | excom 2162 | . . . 4 ⊢ (∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) | |
12 | an12 642 | . . . . . . 7 ⊢ ((𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸))) | |
13 | 12 | exbii 1850 | . . . . . 6 ⊢ (∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸))) |
14 | fvex 6787 | . . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V | |
15 | opeq1 4804 | . . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → 〈𝑓, 𝑠〉 = 〈(𝑠‘𝐺), 𝑠〉) | |
16 | 15 | eqeq2d 2749 | . . . . . . . 8 ⊢ (𝑓 = (𝑠‘𝐺) → (𝑌 = 〈𝑓, 𝑠〉 ↔ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
17 | 16 | anbi1d 630 | . . . . . . 7 ⊢ (𝑓 = (𝑠‘𝐺) → ((𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸) ↔ (𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸))) |
18 | 14, 17 | ceqsexv 3479 | . . . . . 6 ⊢ (∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸)) ↔ (𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸)) |
19 | ancom 461 | . . . . . 6 ⊢ ((𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) | |
20 | 13, 18, 19 | 3bitri 297 | . . . . 5 ⊢ (∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
21 | 20 | exbii 1850 | . . . 4 ⊢ (∃𝑠∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
22 | 11, 21 | bitri 274 | . . 3 ⊢ (∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
23 | elopab 5440 | . . 3 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) | |
24 | df-rex 3070 | . . 3 ⊢ (∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) | |
25 | 22, 23, 24 | 3bitr4i 303 | . 2 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉) |
26 | 10, 25 | bitrdi 287 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ ∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∃wrex 3065 〈cop 4567 class class class wbr 5074 {copab 5136 ‘cfv 6433 ℩crio 7231 lecple 16969 occoc 16970 Atomscatm 37277 LHypclh 37998 LTrncltrn 38115 TEndoctendo 38766 DIsoCcdic 39186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-dic 39187 |
This theorem is referenced by: cdlemn11pre 39224 dihord2pre 39239 |
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