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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1442 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 34073. 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 | fnfund 6651 | . . . . 5 ⊢ (𝜒 → Fun 𝑄) |
5 | opex 5465 | . . . . . . . 8 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ V | |
6 | 5 | snid 4665 | . . . . . . 7 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ {⟨𝑥, (𝐺‘𝑍)⟩} |
7 | elun2 4178 | . . . . . . 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 2833 | . . . . 5 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ 𝑄 |
11 | funopfv 6944 | . . . . 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 33769 | . . 3 ⊢ (𝜃 → (𝑄‘𝑥) = (𝐺‘𝑍)) |
14 | 1, 13 | bnj832 33769 | . 2 ⊢ (𝜂 → (𝑄‘𝑥) = (𝐺‘𝑍)) |
15 | elsni 4646 | . . . 4 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥) | |
16 | 1, 15 | simplbiim 506 | . . 3 ⊢ (𝜂 → 𝑧 = 𝑥) |
17 | 16 | fveq2d 6896 | . 2 ⊢ (𝜂 → (𝑄‘𝑧) = (𝑄‘𝑥)) |
18 | bnj602 33926 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅)) | |
19 | 18 | reseq2d 5982 | . . . . . . 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 33781 | . . . . . . . . . 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 499 | . . . . . . . . . . . 12 ⊢ (𝜓 → 𝑅 FrSe 𝐴) |
26 | 23, 25 | bnj835 33770 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑅 FrSe 𝐴) |
27 | bnj1442.5 | . . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
28 | 27, 23 | bnj1212 33810 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑥 ∈ 𝐴) |
29 | bnj906 33941 | . . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) | |
30 | 26, 28, 29 | syl2anc 585 | . . . . . . . . . 10 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) |
31 | bnj1442.15 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅)) | |
32 | 31 | fndmd 6655 | . . . . . . . . . 10 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) |
33 | 30, 32 | sseqtrrd 4024 | . . . . . . . . 9 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃) |
34 | 4, 22, 33 | bnj1503 33860 | . . . . . . . 8 ⊢ (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
35 | 2, 34 | bnj832 33769 | . . . . . . 7 ⊢ (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
36 | 1, 35 | bnj832 33769 | . . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
37 | 20, 36 | eqtrd 2773 | . . . . 5 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
38 | 16, 37 | opeq12d 4882 | . . . 4 ⊢ (𝜂 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩) |
39 | bnj1442.13 | . . . 4 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ | |
40 | bnj1442.11 | . . . 4 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩ | |
41 | 38, 39, 40 | 3eqtr4g 2798 | . . 3 ⊢ (𝜂 → 𝑊 = 𝑍) |
42 | 41 | fveq2d 6896 | . 2 ⊢ (𝜂 → (𝐺‘𝑊) = (𝐺‘𝑍)) |
43 | 14, 17, 42 | 3eqtr4d 2783 | 1 ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 [wsbc 3778 ∪ cun 3947 ⊆ wss 3949 ∅c0 4323 {csn 4629 ⟨cop 4635 ∪ cuni 4909 class class class wbr 5149 dom cdm 5677 ↾ cres 5679 Fun wfun 6538 Fn wfn 6539 ‘cfv 6544 predc-bnj14 33699 FrSe w-bnj15 33703 trClc-bnj18 33705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-reg 9587 ax-inf2 9636 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-om 7856 df-1o 8466 df-bnj17 33698 df-bnj14 33700 df-bnj13 33702 df-bnj15 33704 df-bnj18 33706 |
This theorem is referenced by: bnj1423 34062 |
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