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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 36468 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)}) |
10 | 9 | eleq2d 2687 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})) |
11 | excom 2042 | . . . 4 ⊢ (∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) | |
12 | an12 838 | . . . . . . 7 ⊢ ((𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸))) | |
13 | 12 | exbii 1774 | . . . . . 6 ⊢ (∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸))) |
14 | fvex 6201 | . . . . . . 7 ⊢ (𝑠‘𝐺) ∈ V | |
15 | opeq1 4402 | . . . . . . . . 9 ⊢ (𝑓 = (𝑠‘𝐺) → 〈𝑓, 𝑠〉 = 〈(𝑠‘𝐺), 𝑠〉) | |
16 | 15 | eqeq2d 2632 | . . . . . . . 8 ⊢ (𝑓 = (𝑠‘𝐺) → (𝑌 = 〈𝑓, 𝑠〉 ↔ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
17 | 16 | anbi1d 741 | . . . . . . 7 ⊢ (𝑓 = (𝑠‘𝐺) → ((𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸) ↔ (𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸))) |
18 | 14, 17 | ceqsexv 3242 | . . . . . 6 ⊢ (∃𝑓(𝑓 = (𝑠‘𝐺) ∧ (𝑌 = 〈𝑓, 𝑠〉 ∧ 𝑠 ∈ 𝐸)) ↔ (𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸)) |
19 | ancom 466 | . . . . . 6 ⊢ ((𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ∧ 𝑠 ∈ 𝐸) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) | |
20 | 13, 18, 19 | 3bitri 286 | . . . . 5 ⊢ (∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ (𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
21 | 20 | exbii 1774 | . . . 4 ⊢ (∃𝑠∃𝑓(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
22 | 11, 21 | bitri 264 | . . 3 ⊢ (∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)) ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) |
23 | elopab 4983 | . . 3 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑓∃𝑠(𝑌 = 〈𝑓, 𝑠〉 ∧ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸))) | |
24 | df-rex 2918 | . . 3 ⊢ (∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉 ↔ ∃𝑠(𝑠 ∈ 𝐸 ∧ 𝑌 = 〈(𝑠‘𝐺), 𝑠〉)) | |
25 | 22, 23, 24 | 3bitr4i 292 | . 2 ⊢ (𝑌 ∈ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} ↔ ∃𝑠 ∈ 𝐸 𝑌 = 〈(𝑠‘𝐺), 𝑠〉) |
26 | 10, 25 | syl6bb 276 | 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 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃wrex 2913 〈cop 4183 class class class wbr 4653 {copab 4712 ‘cfv 5888 ℩crio 6610 lecple 15948 occoc 15949 Atomscatm 34550 LHypclh 35270 LTrncltrn 35387 TEndoctendo 36040 DIsoCcdic 36461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-dic 36462 |
This theorem is referenced by: cdlemn11pre 36499 dihord2pre 36514 |
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