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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 38839 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}) |
10 | 9 | eleq2d 2819 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})) |
11 | excom 2170 | . . . 4 ⊢ (∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) | |
12 | an12 645 | . . . . . . 7 ⊢ ((𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸))) | |
13 | 12 | exbii 1854 | . . . . . 6 ⊢ (∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸))) |
14 | fvex 6690 | . . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V | |
15 | opeq1 4760 | . . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → 〈𝑓, 𝑠〉 = 〈(𝑠‘𝐺), 𝑠〉) | |
16 | 15 | eqeq2d 2750 | . . . . . . . 8 ⊢ (𝑓 = (𝑠‘𝐺) → (𝑌 = 〈𝑓, 𝑠〉 ↔ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
17 | 16 | anbi1d 633 | . . . . . . 7 ⊢ (𝑓 = (𝑠‘𝐺) → ((𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸) ↔ (𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸))) |
18 | 14, 17 | ceqsexv 3446 | . . . . . 6 ⊢ (∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸)) ↔ (𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸)) |
19 | ancom 464 | . . . . . 6 ⊢ ((𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) | |
20 | 13, 18, 19 | 3bitri 300 | . . . . 5 ⊢ (∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
21 | 20 | exbii 1854 | . . . 4 ⊢ (∃𝑠∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
22 | 11, 21 | bitri 278 | . . 3 ⊢ (∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
23 | elopab 5383 | . . 3 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) | |
24 | df-rex 3060 | . . 3 ⊢ (∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) | |
25 | 22, 23, 24 | 3bitr4i 306 | . 2 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉) |
26 | 10, 25 | bitrdi 290 | 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 209 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2114 ∃wrex 3055 〈cop 4523 class class class wbr 5031 {copab 5093 ‘cfv 6340 ℩crio 7129 lecple 16678 occoc 16679 Atomscatm 36923 LHypclh 37644 LTrncltrn 37761 TEndoctendo 38412 DIsoCcdic 38832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-dic 38833 |
This theorem is referenced by: cdlemn11pre 38870 dihord2pre 38885 |
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