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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 38317 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}) |
10 | 9 | eleq2d 2900 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})) |
11 | excom 2169 | . . . 4 ⊢ (∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) | |
12 | an12 643 | . . . . . . 7 ⊢ ((𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸))) | |
13 | 12 | exbii 1848 | . . . . . 6 ⊢ (∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸))) |
14 | fvex 6685 | . . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V | |
15 | opeq1 4805 | . . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → 〈𝑓, 𝑠〉 = 〈(𝑠‘𝐺), 𝑠〉) | |
16 | 15 | eqeq2d 2834 | . . . . . . . 8 ⊢ (𝑓 = (𝑠‘𝐺) → (𝑌 = 〈𝑓, 𝑠〉 ↔ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
17 | 16 | anbi1d 631 | . . . . . . 7 ⊢ (𝑓 = (𝑠‘𝐺) → ((𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸) ↔ (𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸))) |
18 | 14, 17 | ceqsexv 3543 | . . . . . 6 ⊢ (∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸)) ↔ (𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸)) |
19 | ancom 463 | . . . . . 6 ⊢ ((𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) | |
20 | 13, 18, 19 | 3bitri 299 | . . . . 5 ⊢ (∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
21 | 20 | exbii 1848 | . . . 4 ⊢ (∃𝑠∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
22 | 11, 21 | bitri 277 | . . 3 ⊢ (∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
23 | elopab 5416 | . . 3 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) | |
24 | df-rex 3146 | . . 3 ⊢ (∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) | |
25 | 22, 23, 24 | 3bitr4i 305 | . 2 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉) |
26 | 10, 25 | syl6bb 289 | 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 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃wrex 3141 〈cop 4575 class class class wbr 5068 {copab 5130 ‘cfv 6357 ℩crio 7115 lecple 16574 occoc 16575 Atomscatm 36401 LHypclh 37122 LTrncltrn 37239 TEndoctendo 37890 DIsoCcdic 38310 |
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-dic 38311 |
This theorem is referenced by: cdlemn11pre 38348 dihord2pre 38363 |
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