Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1421 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32329. 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 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | fvex 6677 | . . . . 5 ⊢ (𝐺‘𝑍) ∈ V | |
4 | 2, 3 | funsn 6401 | . . . 4 ⊢ Fun {〈𝑥, (𝐺‘𝑍)〉} |
5 | 1, 4 | jctir 523 | . . 3 ⊢ (𝜒 → (Fun 𝑃 ∧ Fun {〈𝑥, (𝐺‘𝑍)〉})) |
6 | bnj1421.15 | . . . . 5 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) | |
7 | 3 | dmsnop 6067 | . . . . . 6 ⊢ dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥} |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜒 → dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥}) |
9 | 6, 8 | ineq12d 4189 | . . . 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 500 | . . . . . . 7 ⊢ (𝜓 → 𝑅 FrSe 𝐴) |
13 | 10, 12 | bnj835 32025 | . . . . . 6 ⊢ (𝜒 → 𝑅 FrSe 𝐴) |
14 | biid 263 | . . . . . . . 8 ⊢ (𝑅 FrSe 𝐴 ↔ 𝑅 FrSe 𝐴) | |
15 | biid 263 | . . . . . . . 8 ⊢ (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) | |
16 | biid 263 | . . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))) | |
17 | biid 263 | . . . . . . . 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 2821 | . . . . . . . 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 32308 | . . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
20 | disjsn 4640 | . . . . . . . 8 ⊢ (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) | |
21 | 20 | ralbii 3165 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
22 | 19, 21 | sylibr 236 | . . . . . 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 32066 | . . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴) |
26 | 23, 25 | bnj1294 32084 | . . . 4 ⊢ (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅) |
27 | 9, 26 | eqtrd 2856 | . . 3 ⊢ (𝜒 → (dom 𝑃 ∩ dom {〈𝑥, (𝐺‘𝑍)〉}) = ∅) |
28 | funun 6394 | . . 3 ⊢ (((Fun 𝑃 ∧ Fun {〈𝑥, (𝐺‘𝑍)〉}) ∧ (dom 𝑃 ∩ dom {〈𝑥, (𝐺‘𝑍)〉}) = ∅) → Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) | |
29 | 5, 27, 28 | syl2anc 586 | . 2 ⊢ (𝜒 → Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) |
30 | bnj1421.12 | . . 3 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
31 | 30 | funeqi 6370 | . 2 ⊢ (Fun 𝑄 ↔ Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) |
32 | 29, 31 | sylibr 236 | 1 ⊢ (𝜒 → Fun 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 [wsbc 3771 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4560 〈cop 4566 ∪ cuni 4831 ∪ ciun 4911 class class class wbr 5058 dom cdm 5549 ↾ cres 5551 Fun wfun 6343 Fn wfn 6344 ‘cfv 6349 predc-bnj14 31953 FrSe w-bnj15 31957 trClc-bnj18 31959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-reg 9050 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-1o 8096 df-bnj17 31952 df-bnj14 31954 df-bnj13 31956 df-bnj15 31958 df-bnj18 31960 df-bnj19 31962 |
This theorem is referenced by: bnj1312 32325 |
Copyright terms: Public domain | W3C validator |