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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1053 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32282. 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 |
---|---|
bnj1053.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj1053.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj1053.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj1053.4 | ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) |
bnj1053.5 | ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) |
bnj1053.6 | ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) |
bnj1053.7 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj1053.8 | ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj1053.9 | ⊢ (𝜂 ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) |
bnj1053.10 | ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂)) |
bnj1053.37 | ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) |
Ref | Expression |
---|---|
bnj1053 | ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1053.1 | . 2 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
2 | bnj1053.2 | . 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
3 | bnj1053.3 | . 2 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
4 | bnj1053.4 | . 2 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) | |
5 | bnj1053.5 | . 2 ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) | |
6 | bnj1053.6 | . 2 ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) | |
7 | bnj1053.7 | . 2 ⊢ 𝐷 = (ω ∖ {∅}) | |
8 | bnj1053.8 | . 2 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
9 | bnj1053.9 | . 2 ⊢ (𝜂 ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) | |
10 | bnj1053.10 | . 2 ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂)) | |
11 | 7 | bnj923 32039 | . . . . . 6 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
12 | nnord 7588 | . . . . . 6 ⊢ (𝑛 ∈ ω → Ord 𝑛) | |
13 | ordfr 6206 | . . . . . 6 ⊢ (Ord 𝑛 → E Fr 𝑛) | |
14 | 11, 12, 13 | 3syl 18 | . . . . 5 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛) |
15 | 3, 14 | bnj769 32033 | . . . 4 ⊢ (𝜒 → E Fr 𝑛) |
16 | 15 | bnj707 32026 | . . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → E Fr 𝑛) |
17 | bnj1053.37 | . . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)) | |
18 | 16, 17 | jca 514 | . 2 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂))) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18 | bnj1052 32247 | 1 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {cab 2799 ∀wral 3138 ∃wrex 3139 Vcvv 3494 [wsbc 3772 ∖ cdif 3933 ⊆ wss 3936 ∅c0 4291 {csn 4567 ∪ ciun 4919 class class class wbr 5066 E cep 5464 Fr wfr 5511 Ord word 6190 suc csuc 6193 Fn wfn 6350 ‘cfv 6355 ωcom 7580 ∧ w-bnj17 31956 predc-bnj14 31958 FrSe w-bnj15 31962 trClc-bnj18 31964 TrFow-bnj19 31966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-fn 6358 df-om 7581 df-bnj17 31957 df-bnj18 31965 |
This theorem is referenced by: (None) |
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