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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj151 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj153 32147. 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 32143 | . . . . . 6 ⊢ 𝜃0 |
13 | 12, 6 | mpbi 232 | . . . . 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 32136 | . . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) |
20 | 14, 15, 16, 17, 18, 19 | bnj149 32142 | . . . . . 6 ⊢ 𝜃1 |
21 | 20, 14 | mpbi 232 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) |
22 | df-eu 2650 | . . . . 5 ⊢ (∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ∧ ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) | |
23 | 13, 21, 22 | sylanbrc 585 | . . . 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 32141 | . . . 4 ⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) |
27 | 23, 26 | mpbir 233 | . . 3 ⊢ 𝜃′ |
28 | sbceq1a 3783 | . . . 4 ⊢ (𝑛 = 1o → (𝜃 ↔ [1o / 𝑛]𝜃)) | |
29 | 28, 25 | syl6bbr 291 | . . 3 ⊢ (𝑛 = 1o → (𝜃 ↔ 𝜃′)) |
30 | 27, 29 | mpbiri 260 | . 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∃*wmo 2616 ∃!weu 2649 ∀wral 3138 [wsbc 3772 ∖ cdif 3933 ∅c0 4291 {csn 4561 〈cop 4567 ∪ ciun 4912 class class class wbr 5059 E cep 5459 suc csuc 6188 Fn wfn 6345 ‘cfv 6350 ωcom 7574 1oc1o 8089 predc-bnj14 31953 FrSe w-bnj15 31957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-1o 8096 df-bnj13 31956 df-bnj15 31958 |
This theorem is referenced by: bnj153 32147 |
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