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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1030 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 31204. 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 |
---|---|
bnj1030.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj1030.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj1030.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj1030.4 | ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) |
bnj1030.5 | ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) |
bnj1030.6 | ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) |
bnj1030.7 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj1030.8 | ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj1030.9 | ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵)) |
bnj1030.10 | ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂)) |
bnj1030.11 | ⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑) |
bnj1030.12 | ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓) |
bnj1030.13 | ⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒) |
bnj1030.14 | ⊢ (𝜃′ ↔ [𝑗 / 𝑖]𝜃) |
bnj1030.15 | ⊢ (𝜏′ ↔ [𝑗 / 𝑖]𝜏) |
bnj1030.16 | ⊢ (𝜁′ ↔ [𝑗 / 𝑖]𝜁) |
bnj1030.17 | ⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂) |
bnj1030.18 | ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′)) |
bnj1030.19 | ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓)) |
Ref | Expression |
---|---|
bnj1030 | ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1030.1 | . 2 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
2 | bnj1030.2 | . 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
3 | bnj1030.3 | . 2 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
4 | bnj1030.4 | . 2 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) | |
5 | bnj1030.5 | . 2 ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) | |
6 | bnj1030.6 | . 2 ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) | |
7 | bnj1030.7 | . 2 ⊢ 𝐷 = (ω ∖ {∅}) | |
8 | bnj1030.8 | . 2 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
9 | 19.23vv 1912 | . . . . 5 ⊢ (∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) | |
10 | 9 | albii 1787 | . . . 4 ⊢ (∀𝑓∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ ∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) |
11 | 19.23v 1911 | . . . 4 ⊢ (∀𝑓(∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) | |
12 | 10, 11 | bitri 264 | . . 3 ⊢ (∀𝑓∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) ↔ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) |
13 | bnj1030.9 | . . . . 5 ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵)) | |
14 | 7 | bnj1071 31171 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
15 | 3, 14 | bnj769 30958 | . . . . . . 7 ⊢ (𝜒 → E Fr 𝑛) |
16 | 15 | bnj707 30951 | . . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → E Fr 𝑛) |
17 | bnj1030.10 | . . . . . . 7 ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂)) | |
18 | bnj1030.17 | . . . . . . 7 ⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂) | |
19 | bnj1030.18 | . . . . . . 7 ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′)) | |
20 | bnj1030.19 | . . . . . . 7 ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓)) | |
21 | 2, 8, 13, 18 | bnj1123 31180 | . . . . . . . . . 10 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)) |
22 | 2, 3, 5, 7, 19, 20, 21 | bnj1118 31178 | . . . . . . . . 9 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵) |
23 | 1, 3, 5 | bnj1097 31175 | . . . . . . . . 9 ⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵) |
24 | 22, 23 | bnj1109 30983 | . . . . . . . 8 ⊢ ∃𝑗(((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘𝑖) ⊆ 𝐵) |
25 | 24, 2, 3 | bnj1093 31174 | . . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖∃𝑗(𝜑0 → (𝑓‘𝑖) ⊆ 𝐵)) |
26 | 13, 17, 18, 19, 20, 25 | bnj1090 31173 | . . . . . 6 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) |
27 | vex 3234 | . . . . . . 7 ⊢ 𝑛 ∈ V | |
28 | 27, 17 | bnj110 31054 | . . . . . 6 ⊢ (( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) → ∀𝑖 ∈ 𝑛 𝜂) |
29 | 16, 26, 28 | syl2anc 694 | . . . . 5 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 𝜂) |
30 | 4, 5, 3, 6, 13, 29, 8 | bnj1121 31179 | . . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) |
31 | 30 | gen2 1763 | . . 3 ⊢ ∀𝑛∀𝑖((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) |
32 | 12, 31 | mpgbi 1765 | . 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵) |
33 | 1, 2, 3, 4, 5, 6, 7, 8, 32 | bnj1034 31164 | 1 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 ∀wal 1521 = wceq 1523 ∃wex 1744 ∈ wcel 2030 {cab 2637 ∀wral 2941 ∃wrex 2942 Vcvv 3231 [wsbc 3468 ∖ cdif 3604 ⊆ wss 3607 ∅c0 3948 {csn 4210 ∪ ciun 4552 class class class wbr 4685 E cep 5057 Fr wfr 5099 dom cdm 5143 suc csuc 5763 Fn wfn 5921 ‘cfv 5926 ωcom 7107 ∧ w-bnj17 30880 predc-bnj14 30882 FrSe w-bnj15 30886 trClc-bnj18 30888 TrFow-bnj19 30890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-tr 4786 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fn 5929 df-fv 5934 df-om 7108 df-bnj17 30881 df-bnj18 30889 df-bnj19 30891 |
This theorem is referenced by: bnj1124 31182 |
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