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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1018 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32563. 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 32516 for a less restrictive version requiring ax-13 2372. (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 32238 | . . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏)) | |
2 | bnj258 32259 | . . . . . . . 8 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏)) | |
3 | bnj1018.29 | . . . . . . . 8 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″) | |
4 | 2, 3 | sylbir 238 | . . . . . . 7 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏) → 𝜒″) |
5 | 4 | ex 416 | . . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (𝜏 → 𝜒″)) |
6 | 5 | eximdv 1924 | . . . . 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 32506 | . . . . 5 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″) |
13 | 6, 12 | syl6ibr 255 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (∃𝑝𝜏 → 𝐺 ∈ 𝐵)) |
14 | 13 | imp 410 | . . 3 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏) → 𝐺 ∈ 𝐵) |
15 | 1, 14 | sylbi 220 | . 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → 𝐺 ∈ 𝐵) |
16 | bnj1019 32332 | . . 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 1145 | . . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ 𝑝) |
19 | bnj1018.26 | . . . . . . 7 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) | |
20 | 19 | bnj1235 32357 | . . . . . 6 ⊢ (𝜒″ → 𝐺 Fn 𝑝) |
21 | fndm 6440 | . . . . . 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 2836 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ dom 𝐺) |
24 | 23 | exlimiv 1937 | . . 3 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ dom 𝐺) |
25 | 16, 24 | sylbir 238 | . 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 32318 | . . 3 ⊢ 𝐺 ∈ V |
30 | vex 3402 | . . . 4 ⊢ 𝑖 ∈ V | |
31 | 30 | sucex 7547 | . . 3 ⊢ suc 𝑖 ∈ V |
32 | 26, 27, 28, 10, 29, 31 | bnj1015 32515 | . 2 ⊢ ((𝐺 ∈ 𝐵 ∧ suc 𝑖 ∈ dom 𝐺) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
33 | 15, 25, 32 | syl2anc 587 | 1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∃wex 1786 ∈ wcel 2114 {cab 2716 ∀wral 3053 ∃wrex 3054 Vcvv 3398 [wsbc 3680 ∖ cdif 3840 ∪ cun 3841 ⊆ wss 3843 ∅c0 4211 {csn 4516 〈cop 4522 ∪ ciun 4881 dom cdm 5525 suc csuc 6174 Fn wfn 6334 ‘cfv 6339 ωcom 7601 ∧ w-bnj17 32237 predc-bnj14 32239 FrSe w-bnj15 32243 trClc-bnj18 32245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7481 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-dm 5535 df-suc 6178 df-iota 6297 df-fn 6342 df-fv 6347 df-bnj17 32238 df-bnj18 32246 |
This theorem is referenced by: bnj1020 32518 |
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