Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 38330 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}) |
10 | 9 | eleq2d 2898 | . 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 6683 | . . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V | |
15 | opeq1 4803 | . . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → 〈𝑓, 𝑠〉 = 〈(𝑠‘𝐺), 𝑠〉) | |
16 | 15 | eqeq2d 2832 | . . . . . . . 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 3541 | . . . . . 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 5414 | . . 3 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) | |
24 | df-rex 3144 | . . 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 3139 〈cop 4573 class class class wbr 5066 {copab 5128 ‘cfv 6355 ℩crio 7113 lecple 16572 occoc 16573 Atomscatm 36414 LHypclh 37135 LTrncltrn 37252 TEndoctendo 37903 DIsoCcdic 38323 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-dic 38324 |
This theorem is referenced by: cdlemn11pre 38361 dihord2pre 38376 |
Copyright terms: Public domain | W3C validator |