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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1442 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj60 35076. 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 6669 | . . . . 5 ⊢ (𝜒 → Fun 𝑄) | 
| 5 | opex 5469 | . . . . . . . 8 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ V | |
| 6 | 5 | snid 4662 | . . . . . . 7 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ {〈𝑥, (𝐺‘𝑍)〉} | 
| 7 | elun2 4183 | . . . . . . 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 2840 | . . . . 5 ⊢ 〈𝑥, (𝐺‘𝑍)〉 ∈ 𝑄 | 
| 11 | funopfv 6958 | . . . . 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 34772 | . . 3 ⊢ (𝜃 → (𝑄‘𝑥) = (𝐺‘𝑍)) | 
| 14 | 1, 13 | bnj832 34772 | . 2 ⊢ (𝜂 → (𝑄‘𝑥) = (𝐺‘𝑍)) | 
| 15 | elsni 4643 | . . . 4 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥) | |
| 16 | 1, 15 | simplbiim 504 | . . 3 ⊢ (𝜂 → 𝑧 = 𝑥) | 
| 17 | 16 | fveq2d 6910 | . 2 ⊢ (𝜂 → (𝑄‘𝑧) = (𝑄‘𝑥)) | 
| 18 | bnj602 34929 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅)) | |
| 19 | 18 | reseq2d 5997 | . . . . . . 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 34784 | . . . . . . . . . 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 497 | . . . . . . . . . . . 12 ⊢ (𝜓 → 𝑅 FrSe 𝐴) | 
| 26 | 23, 25 | bnj835 34773 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑅 FrSe 𝐴) | 
| 27 | bnj1442.5 | . . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
| 28 | 27, 23 | bnj1212 34813 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑥 ∈ 𝐴) | 
| 29 | bnj906 34944 | . . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) | |
| 30 | 26, 28, 29 | syl2anc 584 | . . . . . . . . . 10 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅)) | 
| 31 | bnj1442.15 | . . . . . . . . . . 11 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅)) | |
| 32 | 31 | fndmd 6673 | . . . . . . . . . 10 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) | 
| 33 | 30, 32 | sseqtrrd 4021 | . . . . . . . . 9 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃) | 
| 34 | 4, 22, 33 | bnj1503 34863 | . . . . . . . 8 ⊢ (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) | 
| 35 | 2, 34 | bnj832 34772 | . . . . . . 7 ⊢ (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) | 
| 36 | 1, 35 | bnj832 34772 | . . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) | 
| 37 | 20, 36 | eqtrd 2777 | . . . . 5 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))) | 
| 38 | 16, 37 | opeq12d 4881 | . . . 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 2802 | . . 3 ⊢ (𝜂 → 𝑊 = 𝑍) | 
| 42 | 41 | fveq2d 6910 | . 2 ⊢ (𝜂 → (𝐺‘𝑊) = (𝐺‘𝑍)) | 
| 43 | 14, 17, 42 | 3eqtr4d 2787 | 1 ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 [wsbc 3788 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 {csn 4626 〈cop 4632 ∪ cuni 4907 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 predc-bnj14 34702 FrSe w-bnj15 34706 trClc-bnj18 34708 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-bnj17 34701 df-bnj14 34703 df-bnj13 34705 df-bnj15 34707 df-bnj18 34709 | 
| This theorem is referenced by: bnj1423 35065 | 
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