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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1442 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32709. 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 32416 | . . . . 5 ⊢ (𝜒 → Fun 𝑄) |
5 | opex 5333 | . . . . . . . 8 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ V | |
6 | 5 | snid 4563 | . . . . . . 7 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ {〈𝑥, (𝐺‘𝑍)〉} |
7 | elun2 4077 | . . . . . . 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 2830 | . . . . 5 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ 𝑄 |
11 | funopfv 6742 | . . . . 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 32404 | . . 3 ⊢ (𝜃 → (𝑄‘𝑥) = (𝐺‘𝑍)) |
14 | 1, 13 | bnj832 32404 | . 2 ⊢ (𝜂 → (𝑄‘𝑥) = (𝐺‘𝑍)) |
15 | elsni 4544 | . . . 4 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥) | |
16 | 1, 15 | simplbiim 508 | . . 3 ⊢ (𝜂 → 𝑧 = 𝑥) |
17 | 16 | fveq2d 6699 | . 2 ⊢ (𝜂 → (𝑄‘𝑧) = (𝑄‘𝑥)) |
18 | bnj602 32562 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅)) | |
19 | 18 | reseq2d 5836 | . . . . . . 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 32417 | . . . . . . . . . 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 501 | . . . . . . . . . . . 12 ⊢ (𝜓 → 𝑅 FrSe 𝐴) |
26 | 23, 25 | bnj835 32405 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑅 FrSe 𝐴) |
27 | bnj1442.5 | . . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
28 | 27, 23 | bnj1212 32446 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑥 ∈ 𝐴) |
29 | bnj906 32577 | . . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) | |
30 | 26, 28, 29 | syl2anc 587 | . . . . . . . . . 10 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) |
31 | bnj1442.15 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅)) | |
32 | 31 | fndmd 6461 | . . . . . . . . . 10 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) |
33 | 30, 32 | sseqtrrd 3928 | . . . . . . . . 9 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃) |
34 | 4, 22, 33 | bnj1503 32496 | . . . . . . . 8 ⊢ (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
35 | 2, 34 | bnj832 32404 | . . . . . . 7 ⊢ (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
36 | 1, 35 | bnj832 32404 | . . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
37 | 20, 36 | eqtrd 2771 | . . . . 5 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
38 | 16, 37 | opeq12d 4778 | . . . 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 2796 | . . 3 ⊢ (𝜂 → 𝑊 = 𝑍) |
42 | 41 | fveq2d 6699 | . 2 ⊢ (𝜂 → (𝐺‘𝑊) = (𝐺‘𝑍)) |
43 | 14, 17, 42 | 3eqtr4d 2781 | 1 ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2112 {cab 2714 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 {crab 3055 [wsbc 3683 ∪ cun 3851 ⊆ wss 3853 ∅c0 4223 {csn 4527 〈cop 4533 ∪ cuni 4805 class class class wbr 5039 dom cdm 5536 ↾ cres 5538 Fun wfun 6352 Fn wfn 6353 ‘cfv 6358 predc-bnj14 32333 FrSe w-bnj15 32337 trClc-bnj18 32339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-reg 9186 ax-inf2 9234 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-1o 8180 df-bnj17 32332 df-bnj14 32334 df-bnj13 32336 df-bnj15 32338 df-bnj18 32340 |
This theorem is referenced by: bnj1423 32698 |
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