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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj151 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj153 34894. 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 |
|---|---|
| bnj151.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
| bnj151.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| bnj151.3 | ⊢ 𝐷 = (ω ∖ {∅}) |
| bnj151.4 | ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
| bnj151.5 | ⊢ (𝜏 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜃)) |
| bnj151.6 | ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) |
| bnj151.7 | ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑) |
| bnj151.8 | ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓) |
| bnj151.9 | ⊢ (𝜃′ ↔ [1o / 𝑛]𝜃) |
| bnj151.10 | ⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) |
| bnj151.11 | ⊢ (𝜃1 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) |
| bnj151.12 | ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁) |
| bnj151.13 | ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
| bnj151.14 | ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′) |
| bnj151.15 | ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′) |
| bnj151.16 | ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′) |
| bnj151.17 | ⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) |
| bnj151.18 | ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0) |
| bnj151.19 | ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′) |
| bnj151.20 | ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′) |
| Ref | Expression |
|---|---|
| bnj151 | ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj151.1 | . . . . . . 7 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) | |
| 2 | bnj151.2 | . . . . . . 7 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
| 3 | bnj151.6 | . . . . . . 7 ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) | |
| 4 | bnj151.7 | . . . . . . 7 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑) | |
| 5 | bnj151.8 | . . . . . . 7 ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓) | |
| 6 | bnj151.10 | . . . . . . 7 ⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) | |
| 7 | bnj151.12 | . . . . . . 7 ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁) | |
| 8 | bnj151.13 | . . . . . . 7 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
| 9 | bnj151.14 | . . . . . . 7 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′) | |
| 10 | bnj151.15 | . . . . . . 7 ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′) | |
| 11 | bnj151.16 | . . . . . . 7 ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′) | |
| 12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | bnj150 34890 | . . . . . 6 ⊢ 𝜃0 |
| 13 | 12, 6 | mpbi 230 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) |
| 14 | bnj151.11 | . . . . . . 7 ⊢ (𝜃1 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) | |
| 15 | bnj151.17 | . . . . . . 7 ⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) | |
| 16 | bnj151.18 | . . . . . . 7 ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0) | |
| 17 | bnj151.19 | . . . . . . 7 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′) | |
| 18 | bnj151.20 | . . . . . . 7 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′) | |
| 19 | 1, 4 | bnj118 34883 | . . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
| 20 | 14, 15, 16, 17, 18, 19 | bnj149 34889 | . . . . . 6 ⊢ 𝜃1 |
| 21 | 20, 14 | mpbi 230 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) |
| 22 | df-eu 2569 | . . . . 5 ⊢ (∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ∧ ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) | |
| 23 | 13, 21, 22 | sylanbrc 583 | . . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) |
| 24 | bnj151.4 | . . . . 5 ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) | |
| 25 | bnj151.9 | . . . . 5 ⊢ (𝜃′ ↔ [1o / 𝑛]𝜃) | |
| 26 | 24, 4, 5, 25 | bnj130 34888 | . . . 4 ⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) |
| 27 | 23, 26 | mpbir 231 | . . 3 ⊢ 𝜃′ |
| 28 | sbceq1a 3799 | . . . 4 ⊢ (𝑛 = 1o → (𝜃 ↔ [1o / 𝑛]𝜃)) | |
| 29 | 28, 25 | bitr4di 289 | . . 3 ⊢ (𝑛 = 1o → (𝜃 ↔ 𝜃′)) |
| 30 | 27, 29 | mpbiri 258 | . 2 ⊢ (𝑛 = 1o → 𝜃) |
| 31 | 30 | a1d 25 | 1 ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃*wmo 2538 ∃!weu 2568 ∀wral 3061 [wsbc 3788 ∖ cdif 3948 ∅c0 4333 {csn 4626 〈cop 4632 ∪ ciun 4991 class class class wbr 5143 E cep 5583 suc csuc 6386 Fn wfn 6556 ‘cfv 6561 ωcom 7887 1oc1o 8499 predc-bnj14 34702 FrSe w-bnj15 34706 |
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 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-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1o 8506 df-bnj13 34705 df-bnj15 34707 |
| This theorem is referenced by: bnj153 34894 |
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