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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1020 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 33679. 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 33448 | . . 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 33626 | . . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″) |
19 | 4, 6, 7, 14, 18 | bnj1001 33628 | . . . . 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 33629 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖)) |
21 | 20 | exlimiv 1934 | . . 3 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖)) |
22 | 1, 21 | sylbir 234 | . 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 33633 | . 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
25 | 22, 24 | sstrd 3955 | 1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ∀wral 3061 ∃wrex 3070 [wsbc 3740 ∖ cdif 3908 ∪ cun 3909 ⊆ wss 3911 ∅c0 4283 {csn 4587 ⟨cop 4593 ∪ ciun 4955 suc csuc 6320 Fn wfn 6492 ‘cfv 6497 ωcom 7803 ∧ w-bnj17 33355 predc-bnj14 33357 FrSe w-bnj15 33361 trClc-bnj18 33363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 ax-reg 9533 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-om 7804 df-bnj17 33356 df-bnj14 33358 df-bnj13 33360 df-bnj15 33362 df-bnj18 33364 |
This theorem is referenced by: bnj907 33636 |
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