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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1020 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 34964. 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 |
---|---|
bnj1020.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj1020.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj1020.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj1020.4 | ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅))) |
bnj1020.5 | ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) |
bnj1020.6 | ⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖))) |
bnj1020.7 | ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) |
bnj1020.8 | ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓) |
bnj1020.9 | ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) |
bnj1020.10 | ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) |
bnj1020.11 | ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′) |
bnj1020.12 | ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) |
bnj1020.13 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj1020.14 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj1020.15 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) |
bnj1020.16 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) |
bnj1020.26 | ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) |
Ref | Expression |
---|---|
bnj1020 | ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1019 34733 | . . 3 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) | |
2 | bnj1020.1 | . . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
3 | bnj1020.2 | . . . . 5 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
4 | bnj1020.3 | . . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
5 | bnj1020.4 | . . . . 5 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅))) | |
6 | bnj1020.5 | . . . . 5 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) | |
7 | bnj1020.6 | . . . . 5 ⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖))) | |
8 | bnj1020.7 | . . . . 5 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) | |
9 | bnj1020.8 | . . . . 5 ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓) | |
10 | bnj1020.9 | . . . . 5 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) | |
11 | bnj1020.10 | . . . . 5 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) | |
12 | bnj1020.11 | . . . . 5 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′) | |
13 | bnj1020.12 | . . . . 5 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | |
14 | bnj1020.13 | . . . . 5 ⊢ 𝐷 = (ω ∖ {∅}) | |
15 | bnj1020.15 | . . . . 5 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
16 | bnj1020.16 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
17 | bnj1020.14 | . . . . . . 7 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
18 | 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 15, 16 | bnj998 34911 | . . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″) |
19 | 4, 6, 7, 14, 18 | bnj1001 34913 | . . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝)) |
20 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19 | bnj1006 34914 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖)) |
21 | 20 | exlimiv 1926 | . . 3 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖)) |
22 | 1, 21 | sylbir 235 | . 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖)) |
23 | bnj1020.26 | . . 3 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) | |
24 | 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 15, 16, 23, 18, 19 | bnj1018 34918 | . 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
25 | 22, 24 | sstrd 4006 | 1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1085 = wceq 1535 ∃wex 1774 ∈ wcel 2104 {cab 2710 ∀wral 3057 ∃wrex 3066 [wsbc 3791 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 {csn 4631 〈cop 4637 ∪ ciun 4999 suc csuc 6383 Fn wfn 6554 ‘cfv 6559 ωcom 7881 ∧ w-bnj17 34640 predc-bnj14 34642 FrSe w-bnj15 34646 trClc-bnj18 34648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7748 ax-reg 9624 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4916 df-iun 5001 df-br 5151 df-opab 5213 df-tr 5268 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-ord 6384 df-on 6385 df-lim 6386 df-suc 6387 df-iota 6511 df-fun 6561 df-fn 6562 df-fv 6567 df-om 7882 df-bnj17 34641 df-bnj14 34643 df-bnj13 34645 df-bnj15 34647 df-bnj18 34649 |
This theorem is referenced by: bnj907 34921 |
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