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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1421 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32615. 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 3402 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | fvex 6689 | . . . . 5 ⊢ (𝐺‘𝑍) ∈ V | |
4 | 2, 3 | funsn 6392 | . . . 4 ⊢ Fun {〈𝑥, (𝐺‘𝑍)〉} |
5 | 1, 4 | jctir 524 | . . 3 ⊢ (𝜒 → (Fun 𝑃 ∧ Fun {〈𝑥, (𝐺‘𝑍)〉})) |
6 | bnj1421.15 | . . . . 5 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅)) | |
7 | 3 | dmsnop 6048 | . . . . . 6 ⊢ dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥} |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜒 → dom {〈𝑥, (𝐺‘𝑍)〉} = {𝑥}) |
9 | 6, 8 | ineq12d 4104 | . . . 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 501 | . . . . . . 7 ⊢ (𝜓 → 𝑅 FrSe 𝐴) |
13 | 10, 12 | bnj835 32311 | . . . . . 6 ⊢ (𝜒 → 𝑅 FrSe 𝐴) |
14 | biid 264 | . . . . . . . 8 ⊢ (𝑅 FrSe 𝐴 ↔ 𝑅 FrSe 𝐴) | |
15 | biid 264 | . . . . . . . 8 ⊢ (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) | |
16 | biid 264 | . . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧 ∈ 𝐴 (𝑧𝑅𝑥 → [𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))) | |
17 | biid 264 | . . . . . . . 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 2738 | . . . . . . . 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 32594 | . . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
20 | disjsn 4602 | . . . . . . . 8 ⊢ (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) | |
21 | 20 | ralbii 3080 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) |
22 | 19, 21 | sylibr 237 | . . . . . 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 32352 | . . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴) |
26 | 23, 25 | bnj1294 32370 | . . . 4 ⊢ (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅) |
27 | 9, 26 | eqtrd 2773 | . . 3 ⊢ (𝜒 → (dom 𝑃 ∩ dom {〈𝑥, (𝐺‘𝑍)〉}) = ∅) |
28 | funun 6385 | . . 3 ⊢ (((Fun 𝑃 ∧ Fun {〈𝑥, (𝐺‘𝑍)〉}) ∧ (dom 𝑃 ∩ dom {〈𝑥, (𝐺‘𝑍)〉}) = ∅) → Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) | |
29 | 5, 27, 28 | syl2anc 587 | . 2 ⊢ (𝜒 → Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) |
30 | bnj1421.12 | . . 3 ⊢ 𝑄 = (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉}) | |
31 | 30 | funeqi 6360 | . 2 ⊢ (Fun 𝑄 ↔ Fun (𝑃 ∪ {〈𝑥, (𝐺‘𝑍)〉})) |
32 | 29, 31 | sylibr 237 | 1 ⊢ (𝜒 → Fun 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∃wex 1786 ∈ wcel 2114 {cab 2716 ≠ wne 2934 ∀wral 3053 ∃wrex 3054 {crab 3057 [wsbc 3680 ∪ cun 3841 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 {csn 4516 〈cop 4522 ∪ cuni 4796 ∪ ciun 4881 class class class wbr 5030 dom cdm 5525 ↾ cres 5527 Fun wfun 6333 Fn wfn 6334 ‘cfv 6339 predc-bnj14 32239 FrSe w-bnj15 32243 trClc-bnj18 32245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-reg 9131 ax-inf2 9179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7602 df-1o 8133 df-bnj17 32238 df-bnj14 32240 df-bnj13 32242 df-bnj15 32244 df-bnj18 32246 df-bnj19 32248 |
This theorem is referenced by: bnj1312 32611 |
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