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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1421 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 34821. 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 |
---|---|
bnj1421.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1421.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1421.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1421.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1421.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1421.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1421.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1421.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1421.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1421.10 | ⊢ 𝑃 = ∪ 𝐻 |
bnj1421.11 | ⊢ 𝑍 = 〈𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1421.12 | ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) |
bnj1421.13 | ⊢ (𝜒 → Fun 𝑃) |
bnj1421.14 | ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) |
bnj1421.15 | ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) |
Ref | Expression |
---|---|
bnj1421 | ⊢ (𝜒 → Fun 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1421.13 | . . . 4 ⊢ (𝜒 → Fun 𝑃) | |
2 | vex 3465 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | fvex 6909 | . . . . 5 ⊢ (𝐺‘𝑍) ∈ V | |
4 | 2, 3 | funsn 6607 | . . . 4 ⊢ Fun {〈𝑥, (𝐺‘𝑍)〉} |
5 | 1, 4 | jctir 519 | . . 3 ⊢ (𝜒 → (Fun 𝑃 ∧ Fun {〈𝑥, (𝐺‘𝑍)〉})) |
6 | bnj1421.15 | . . . . 5 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) | |
7 | 3 | dmsnop 6222 | . . . . . 6 ⊢ dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥} |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜒 → dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥}) |
9 | 6, 8 | ineq12d 4211 | . . . 4 ⊢ (𝜒 → (dom 𝑃 ∩ dom {〈𝑥, (𝐺‘𝑍)〉}) = ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥})) |
10 | bnj1421.7 | . . . . . . 7 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) | |
11 | bnj1421.6 | . . . . . . . 8 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) | |
12 | 11 | simplbi 496 | . . . . . . 7 ⊢ (𝜓 → 𝑅 FrSe 𝐴) |
13 | 10, 12 | bnj835 34518 | . . . . . 6 ⊢ (𝜒 → 𝑅 FrSe 𝐴) |
14 | biid 260 | . . . . . . . 8 ⊢ (𝑅 FrSe 𝐴 ↔ 𝑅 FrSe 𝐴) | |
15 | biid 260 | . . . . . . . 8 ⊢ (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) | |
16 | biid 260 | . . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))) | |
17 | biid 260 | . . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)))) | |
18 | eqid 2725 | . . . . . . . 8 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑧 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑧, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑧 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑧, 𝐴, 𝑅)) | |
19 | 14, 15, 16, 17, 18 | bnj1417 34800 | . . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
20 | disjsn 4717 | . . . . . . . 8 ⊢ (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) | |
21 | 20 | ralbii 3082 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
22 | 19, 21 | sylibr 233 | . . . . . 6 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅) |
23 | 13, 22 | syl 17 | . . . . 5 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅) |
24 | bnj1421.5 | . . . . . 6 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
25 | 24, 10 | bnj1212 34558 | . . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴) |
26 | 23, 25 | bnj1294 34576 | . . . 4 ⊢ (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅) |
27 | 9, 26 | eqtrd 2765 | . . 3 ⊢ (𝜒 → (dom 𝑃 ∩ dom {〈𝑥, (𝐺‘𝑍)〉}) = ∅) |
28 | funun 6600 | . . 3 ⊢ (((Fun 𝑃 ∧ Fun {〈𝑥, (𝐺‘𝑍)〉}) ∧ (dom 𝑃 ∩ dom {〈𝑥, (𝐺‘𝑍)〉}) = ∅) → Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) | |
29 | 5, 27, 28 | syl2anc 582 | . 2 ⊢ (𝜒 → Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) |
30 | bnj1421.12 | . . 3 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
31 | 30 | funeqi 6575 | . 2 ⊢ (Fun 𝑄 ↔ Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) |
32 | 29, 31 | sylibr 233 | 1 ⊢ (𝜒 → Fun 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2702 ≠ wne 2929 ∀wral 3050 ∃wrex 3059 {crab 3418 [wsbc 3773 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 {csn 4630 〈cop 4636 ∪ cuni 4909 ∪ ciun 4997 class class class wbr 5149 dom cdm 5678 ↾ cres 5680 Fun wfun 6543 Fn wfn 6544 ‘cfv 6549 predc-bnj14 34447 FrSe w-bnj15 34451 trClc-bnj18 34453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-reg 9617 ax-inf2 9666 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-om 7872 df-1o 8487 df-bnj17 34446 df-bnj14 34448 df-bnj13 34450 df-bnj15 34452 df-bnj18 34454 df-bnj19 34456 |
This theorem is referenced by: bnj1312 34817 |
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