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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1442 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32336. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1442.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1442.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1442.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1442.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1442.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1442.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1442.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1442.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1442.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1442.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1442.11 | ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1442.12 | ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
bnj1442.13 | ⊢ 𝑊 = 〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 |
bnj1442.14 | ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) |
bnj1442.15 | ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅)) |
bnj1442.16 | ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
bnj1442.17 | ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸)) |
bnj1442.18 | ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥})) |
Ref | Expression |
---|---|
bnj1442 | ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1442.18 | . . 3 ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥})) | |
2 | bnj1442.17 | . . . 4 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸)) | |
3 | bnj1442.16 | . . . . . 6 ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) | |
4 | 3 | bnj930 32043 | . . . . 5 ⊢ (𝜒 → Fun 𝑄) |
5 | opex 5358 | . . . . . . . 8 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ V | |
6 | 5 | snid 4603 | . . . . . . 7 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ {〈𝑥, (𝐺‘𝑍)〉} |
7 | elun2 4155 | . . . . . . 7 ⊢ (〈𝑥, (𝐺‘𝑍)〉 ∈ {〈𝑥, (𝐺‘𝑍)〉} → 〈𝑥, (𝐺‘𝑍)〉 ∈ (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
9 | bnj1442.12 | . . . . . 6 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
10 | 8, 9 | eleqtrri 2914 | . . . . 5 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ 𝑄 |
11 | funopfv 6719 | . . . . 5 ⊢ (Fun 𝑄 → (〈𝑥, (𝐺‘𝑍)〉 ∈ 𝑄 → (𝑄‘𝑥) = (𝐺‘𝑍))) | |
12 | 4, 10, 11 | mpisyl 21 | . . . 4 ⊢ (𝜒 → (𝑄‘𝑥) = (𝐺‘𝑍)) |
13 | 2, 12 | bnj832 32031 | . . 3 ⊢ (𝜃 → (𝑄‘𝑥) = (𝐺‘𝑍)) |
14 | 1, 13 | bnj832 32031 | . 2 ⊢ (𝜂 → (𝑄‘𝑥) = (𝐺‘𝑍)) |
15 | elsni 4586 | . . . 4 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥) | |
16 | 1, 15 | simplbiim 507 | . . 3 ⊢ (𝜂 → 𝑧 = 𝑥) |
17 | 16 | fveq2d 6676 | . 2 ⊢ (𝜂 → (𝑄‘𝑧) = (𝑄‘𝑥)) |
18 | bnj602 32189 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅)) | |
19 | 18 | reseq2d 5855 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑥, 𝐴, 𝑅))) |
20 | 16, 19 | syl 17 | . . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑥, 𝐴, 𝑅))) |
21 | 9 | bnj931 32044 | . . . . . . . . . 10 ⊢ 𝑃 ⊆ 𝑄 |
22 | 21 | a1i 11 | . . . . . . . . 9 ⊢ (𝜒 → 𝑃 ⊆ 𝑄) |
23 | bnj1442.7 | . . . . . . . . . . . 12 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) | |
24 | bnj1442.6 | . . . . . . . . . . . . 13 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) | |
25 | 24 | simplbi 500 | . . . . . . . . . . . 12 ⊢ (𝜓 → 𝑅 FrSe 𝐴) |
26 | 23, 25 | bnj835 32032 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑅 FrSe 𝐴) |
27 | bnj1442.5 | . . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
28 | 27, 23 | bnj1212 32073 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑥 ∈ 𝐴) |
29 | bnj906 32204 | . . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) | |
30 | 26, 28, 29 | syl2anc 586 | . . . . . . . . . 10 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) |
31 | bnj1442.15 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅)) | |
32 | fndm 6457 | . . . . . . . . . . 11 ⊢ (𝑃 Fn trCl(𝑥, 𝐴, 𝑅) → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) | |
33 | 31, 32 | syl 17 | . . . . . . . . . 10 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) |
34 | 30, 33 | sseqtrrd 4010 | . . . . . . . . 9 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃) |
35 | 4, 22, 34 | bnj1503 32123 | . . . . . . . 8 ⊢ (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
36 | 2, 35 | bnj832 32031 | . . . . . . 7 ⊢ (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
37 | 1, 36 | bnj832 32031 | . . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
38 | 20, 37 | eqtrd 2858 | . . . . 5 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
39 | 16, 38 | opeq12d 4813 | . . . 4 ⊢ (𝜂 → 〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
40 | bnj1442.13 | . . . 4 ⊢ 𝑊 = 〈𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))〉 | |
41 | bnj1442.11 | . . . 4 ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
42 | 39, 40, 41 | 3eqtr4g 2883 | . . 3 ⊢ (𝜂 → 𝑊 = 𝑍) |
43 | 42 | fveq2d 6676 | . 2 ⊢ (𝜂 → (𝐺‘𝑊) = (𝐺‘𝑍)) |
44 | 14, 17, 43 | 3eqtr4d 2868 | 1 ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 [wsbc 3774 ∪ cun 3936 ⊆ wss 3938 ∅c0 4293 {csn 4569 〈cop 4575 ∪ cuni 4840 class class class wbr 5068 dom cdm 5557 ↾ cres 5559 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 predc-bnj14 31960 FrSe w-bnj15 31964 trClc-bnj18 31966 |
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 ax-reg 9058 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 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-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 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-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 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-om 7583 df-1o 8104 df-bnj17 31959 df-bnj14 31961 df-bnj13 31963 df-bnj15 31965 df-bnj18 31967 |
This theorem is referenced by: bnj1423 32325 |
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