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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1018 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 35002. 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). See bnj1018g 34955 for a less restrictive version requiring ax-13 2374. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1018.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj1018.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj1018.3 | ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
bnj1018.4 | ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅))) |
bnj1018.5 | ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) |
bnj1018.7 | ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑) |
bnj1018.8 | ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓) |
bnj1018.9 | ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) |
bnj1018.10 | ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′) |
bnj1018.11 | ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′) |
bnj1018.12 | ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) |
bnj1018.13 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj1018.14 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj1018.15 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) |
bnj1018.16 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) |
bnj1018.26 | ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) |
bnj1018.29 | ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″) |
bnj1018.30 | ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝)) |
Ref | Expression |
---|---|
bnj1018 | ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bnj17 34679 | . . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏)) | |
2 | bnj258 34700 | . . . . . . . 8 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏)) | |
3 | bnj1018.29 | . . . . . . . 8 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″) | |
4 | 2, 3 | sylbir 235 | . . . . . . 7 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏) → 𝜒″) |
5 | 4 | ex 412 | . . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (𝜏 → 𝜒″)) |
6 | 5 | eximdv 1914 | . . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (∃𝑝𝜏 → ∃𝑝𝜒″)) |
7 | bnj1018.3 | . . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | |
8 | bnj1018.9 | . . . . . 6 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒) | |
9 | bnj1018.12 | . . . . . 6 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′) | |
10 | bnj1018.14 | . . . . . 6 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
11 | bnj1018.16 | . . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
12 | 7, 8, 9, 10, 11 | bnj985v 34945 | . . . . 5 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″) |
13 | 6, 12 | imbitrrdi 252 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (∃𝑝𝜏 → 𝐺 ∈ 𝐵)) |
14 | 13 | imp 406 | . . 3 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏) → 𝐺 ∈ 𝐵) |
15 | 1, 14 | sylbi 217 | . 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → 𝐺 ∈ 𝐵) |
16 | bnj1019 34771 | . . 3 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) | |
17 | bnj1018.30 | . . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝)) | |
18 | 17 | simp3d 1143 | . . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ 𝑝) |
19 | bnj1018.26 | . . . . . . 7 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) | |
20 | 19 | bnj1235 34796 | . . . . . 6 ⊢ (𝜒″ → 𝐺 Fn 𝑝) |
21 | fndm 6671 | . . . . . 6 ⊢ (𝐺 Fn 𝑝 → dom 𝐺 = 𝑝) | |
22 | 3, 20, 21 | 3syl 18 | . . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → dom 𝐺 = 𝑝) |
23 | 18, 22 | eleqtrrd 2841 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ dom 𝐺) |
24 | 23 | exlimiv 1927 | . . 3 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ dom 𝐺) |
25 | 16, 24 | sylbir 235 | . 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → suc 𝑖 ∈ dom 𝐺) |
26 | bnj1018.1 | . . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
27 | bnj1018.2 | . . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
28 | bnj1018.13 | . . 3 ⊢ 𝐷 = (ω ∖ {∅}) | |
29 | 11 | bnj918 34758 | . . 3 ⊢ 𝐺 ∈ V |
30 | vex 3481 | . . . 4 ⊢ 𝑖 ∈ V | |
31 | 30 | sucex 7825 | . . 3 ⊢ suc 𝑖 ∈ V |
32 | 26, 27, 28, 10, 29, 31 | bnj1015 34954 | . 2 ⊢ ((𝐺 ∈ 𝐵 ∧ suc 𝑖 ∈ dom 𝐺) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
33 | 15, 25, 32 | syl2anc 584 | 1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∃wex 1775 ∈ wcel 2105 {cab 2711 ∀wral 3058 ∃wrex 3067 Vcvv 3477 [wsbc 3790 ∖ cdif 3959 ∪ cun 3960 ⊆ wss 3962 ∅c0 4338 {csn 4630 〈cop 4636 ∪ ciun 4995 dom cdm 5688 suc csuc 6387 Fn wfn 6557 ‘cfv 6562 ωcom 7886 ∧ w-bnj17 34678 predc-bnj14 34680 FrSe w-bnj15 34684 trClc-bnj18 34686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-dm 5698 df-suc 6391 df-iota 6515 df-fn 6565 df-fv 6570 df-bnj17 34679 df-bnj18 34687 |
This theorem is referenced by: bnj1020 34957 |
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