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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1442 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 30384. 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 30094 | . . . . 5 ⊢ (𝜒 → Fun 𝑄) |
5 | opex 4859 | . . . . . . . 8 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ V | |
6 | 5 | snid 4155 | . . . . . . 7 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ {〈𝑥, (𝐺‘𝑍)〉} |
7 | elun2 3743 | . . . . . . 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 2687 | . . . . 5 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ 𝑄 |
11 | funopfv 6145 | . . . . 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 30082 | . . 3 ⊢ (𝜃 → (𝑄‘𝑥) = (𝐺‘𝑍)) |
14 | 1, 13 | bnj832 30082 | . 2 ⊢ (𝜂 → (𝑄‘𝑥) = (𝐺‘𝑍)) |
15 | elsni 4142 | . . . 4 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥) | |
16 | 1, 15 | simplbiim 657 | . . 3 ⊢ (𝜂 → 𝑧 = 𝑥) |
17 | 16 | fveq2d 6107 | . 2 ⊢ (𝜂 → (𝑄‘𝑧) = (𝑄‘𝑥)) |
18 | bnj602 30239 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅)) | |
19 | 18 | reseq2d 5317 | . . . . . . 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 30095 | . . . . . . . . . 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 475 | . . . . . . . . . . . 12 ⊢ (𝜓 → 𝑅 FrSe 𝐴) |
26 | 23, 25 | bnj835 30083 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑅 FrSe 𝐴) |
27 | bnj1442.5 | . . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
28 | 27, 23 | bnj1212 30124 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑥 ∈ 𝐴) |
29 | bnj906 30254 | . . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) | |
30 | 26, 28, 29 | syl2anc 691 | . . . . . . . . . 10 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) |
31 | bnj1442.15 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅)) | |
32 | fndm 5904 | . . . . . . . . . . 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 | sseqtr4d 3605 | . . . . . . . . 9 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃) |
35 | 4, 22, 34 | bnj1503 30173 | . . . . . . . 8 ⊢ (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
36 | 2, 35 | bnj832 30082 | . . . . . . 7 ⊢ (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
37 | 1, 36 | bnj832 30082 | . . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
38 | 20, 37 | eqtrd 2644 | . . . . 5 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) |
39 | 16, 38 | opeq12d 4348 | . . . 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 2669 | . . 3 ⊢ (𝜂 → 𝑊 = 𝑍) |
43 | 42 | fveq2d 6107 | . 2 ⊢ (𝜂 → (𝐺‘𝑊) = (𝐺‘𝑍)) |
44 | 14, 17, 43 | 3eqtr4d 2654 | 1 ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 [wsbc 3402 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {csn 4125 〈cop 4131 ∪ cuni 4372 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 predc-bnj14 30007 FrSe w-bnj15 30011 trClc-bnj18 30013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-bnj17 30006 df-bnj14 30008 df-bnj13 30010 df-bnj15 30012 df-bnj18 30014 |
This theorem is referenced by: bnj1423 30373 |
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