|   | Mathbox for Jonathan Ben-Naim | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj998 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj69 35024. 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 | 
|---|---|
| bnj998.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| bnj998.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | 
| bnj998.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | 
| bnj998.4 | ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅))) | 
| bnj998.5 | ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) | 
| bnj998.7 | ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) | 
| bnj998.8 | ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓) | 
| bnj998.9 | ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) | 
| bnj998.10 | ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) | 
| bnj998.11 | ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′) | 
| bnj998.12 | ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | 
| bnj998.13 | ⊢ 𝐷 = (ω ∖ {∅}) | 
| bnj998.14 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | 
| bnj998.15 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | 
| bnj998.16 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | 
| Ref | Expression | 
|---|---|
| bnj998 | ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj998.4 | . . . . . 6 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅))) | |
| 2 | bnj253 34718 | . . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅))) | |
| 3 | 2 | simp1bi 1146 | . . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) | 
| 4 | 1, 3 | sylbi 217 | . . . . 5 ⊢ (𝜃 → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) | 
| 5 | 4 | bnj705 34767 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)) | 
| 6 | bnj643 34763 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒) | |
| 7 | bnj998.5 | . . . . . 6 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) | |
| 8 | 3simpc 1151 | . . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) | |
| 9 | 7, 8 | sylbi 217 | . . . . 5 ⊢ (𝜏 → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) | 
| 10 | 9 | bnj707 34769 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) | 
| 11 | bnj255 34719 | . . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜒 ∧ (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))) | |
| 12 | 5, 6, 10, 11 | syl3anbrc 1344 | . . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) | 
| 13 | bnj252 34717 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))) | |
| 14 | 12, 13 | sylib 218 | . 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))) | 
| 15 | bnj998.1 | . . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
| 16 | bnj998.2 | . . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
| 17 | bnj998.3 | . . 3 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
| 18 | bnj998.7 | . . 3 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) | |
| 19 | bnj998.8 | . . 3 ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓) | |
| 20 | bnj998.9 | . . 3 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) | |
| 21 | bnj998.10 | . . 3 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) | |
| 22 | bnj998.11 | . . 3 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′) | |
| 23 | bnj998.12 | . . 3 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | |
| 24 | bnj998.13 | . . 3 ⊢ 𝐷 = (ω ∖ {∅}) | |
| 25 | bnj998.14 | . . 3 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
| 26 | bnj998.15 | . . 3 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
| 27 | bnj998.16 | . . 3 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
| 28 | biid 261 | . . 3 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
| 29 | biid 261 | . . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛) ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛)) | |
| 30 | 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 | bnj910 34962 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜒″) | 
| 31 | 14, 30 | syl 17 | 1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 [wsbc 3788 ∖ cdif 3948 ∪ cun 3949 ∅c0 4333 {csn 4626 〈cop 4632 ∪ ciun 4991 suc csuc 6386 Fn wfn 6556 ‘cfv 6561 ωcom 7887 ∧ w-bnj17 34700 predc-bnj14 34702 FrSe w-bnj15 34706 trClc-bnj18 34708 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-reg 9632 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-om 7888 df-bnj17 34701 df-bnj14 34703 df-bnj13 34705 df-bnj15 34707 | 
| This theorem is referenced by: bnj1020 34979 | 
| Copyright terms: Public domain | W3C validator |