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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1421 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32231. 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 3495 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | fvex 6676 | . . . . 5 ⊢ (𝐺‘𝑍) ∈ V | |
4 | 2, 3 | funsn 6400 | . . . 4 ⊢ Fun {〈𝑥, (𝐺‘𝑍)〉} |
5 | 1, 4 | jctir 521 | . . 3 ⊢ (𝜒 → (Fun 𝑃 ∧ Fun {〈𝑥, (𝐺‘𝑍)〉})) |
6 | bnj1421.15 | . . . . 5 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) | |
7 | 3 | dmsnop 6066 | . . . . . 6 ⊢ dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥} |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜒 → dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥}) |
9 | 6, 8 | ineq12d 4187 | . . . 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 498 | . . . . . . 7 ⊢ (𝜓 → 𝑅 FrSe 𝐴) |
13 | 10, 12 | bnj835 31929 | . . . . . 6 ⊢ (𝜒 → 𝑅 FrSe 𝐴) |
14 | biid 262 | . . . . . . . 8 ⊢ (𝑅 FrSe 𝐴 ↔ 𝑅 FrSe 𝐴) | |
15 | biid 262 | . . . . . . . 8 ⊢ (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) | |
16 | biid 262 | . . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))) | |
17 | biid 262 | . . . . . . . 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 2818 | . . . . . . . 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 32210 | . . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
20 | disjsn 4639 | . . . . . . . 8 ⊢ (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) | |
21 | 20 | ralbii 3162 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
22 | 19, 21 | sylibr 235 | . . . . . 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 31970 | . . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴) |
26 | 23, 25 | bnj1294 31988 | . . . 4 ⊢ (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅) |
27 | 9, 26 | eqtrd 2853 | . . 3 ⊢ (𝜒 → (dom 𝑃 ∩ dom {〈𝑥, (𝐺‘𝑍)〉}) = ∅) |
28 | funun 6393 | . . 3 ⊢ (((Fun 𝑃 ∧ Fun {〈𝑥, (𝐺‘𝑍)〉}) ∧ (dom 𝑃 ∩ dom {〈𝑥, (𝐺‘𝑍)〉}) = ∅) → Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) | |
29 | 5, 27, 28 | syl2anc 584 | . 2 ⊢ (𝜒 → Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) |
30 | bnj1421.12 | . . 3 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
31 | 30 | funeqi 6369 | . 2 ⊢ (Fun 𝑄 ↔ Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) |
32 | 29, 31 | sylibr 235 | 1 ⊢ (𝜒 → Fun 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∃wex 1771 ∈ wcel 2105 {cab 2796 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 {crab 3139 [wsbc 3769 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 {csn 4557 〈cop 4563 ∪ cuni 4830 ∪ ciun 4910 class class class wbr 5057 dom cdm 5548 ↾ cres 5550 Fun wfun 6342 Fn wfn 6343 ‘cfv 6348 predc-bnj14 31857 FrSe w-bnj15 31861 trClc-bnj18 31863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-reg 9044 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-1o 8091 df-bnj17 31856 df-bnj14 31858 df-bnj13 31860 df-bnj15 31862 df-bnj18 31864 df-bnj19 31866 |
This theorem is referenced by: bnj1312 32227 |
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