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Theorem bnj1423 31636
Description: Technical lemma for bnj60 31647. 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.)
Hypotheses
Ref Expression
bnj1423.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1423.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1423.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1423.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1423.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1423.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1423.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1423.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1423.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1423.10 𝑃 = 𝐻
bnj1423.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1423.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1423.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1423.14 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1423.15 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1423.16 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj1423 (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥,𝑦,𝑧   𝐵,𝑓   𝑦,𝐷   𝐸,𝑑,𝑓,𝑦   𝐺,𝑑,𝑓,𝑥,𝑦,𝑧   𝑅,𝑑,𝑓,𝑥,𝑦,𝑧   𝑧,𝑌   𝜒,𝑧   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐸(𝑥,𝑧)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1423
StepHypRef Expression
1 bnj1423.1 . . . 4 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2 bnj1423.2 . . . 4 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3 bnj1423.3 . . . 4 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
4 bnj1423.4 . . . 4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5 bnj1423.5 . . . 4 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
6 bnj1423.6 . . . 4 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
7 bnj1423.7 . . . 4 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
8 bnj1423.8 . . . 4 (𝜏′[𝑦 / 𝑥]𝜏)
9 bnj1423.9 . . . 4 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
10 bnj1423.10 . . . 4 𝑃 = 𝐻
11 bnj1423.11 . . . 4 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
12 bnj1423.12 . . . 4 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
13 bnj1423.13 . . . 4 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
14 bnj1423.14 . . . 4 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
15 bnj1423.15 . . . 4 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
16 bnj1423.16 . . . 4 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
17 biid 253 . . . 4 ((𝜒𝑧𝐸) ↔ (𝜒𝑧𝐸))
18 biid 253 . . . 4 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) ↔ ((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}))
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18bnj1442 31634 . . 3 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → (𝑄𝑧) = (𝐺𝑊))
20 biid 253 . . . 4 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
21 biid 253 . . . 4 ((((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓𝐻𝑧 ∈ dom 𝑓) ↔ (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓𝐻𝑧 ∈ dom 𝑓))
22 biid 253 . . . 4 (((((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓𝐻𝑧 ∈ dom 𝑓) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ ((((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓𝐻𝑧 ∈ dom 𝑓) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
23 biid 253 . . . 4 ((((((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓𝐻𝑧 ∈ dom 𝑓) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ∧ 𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)) ↔ (((((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ 𝑓𝐻𝑧 ∈ dom 𝑓) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ∧ 𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
24 eqid 2799 . . . 4 𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩
251, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24bnj1450 31635 . . 3 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → (𝑄𝑧) = (𝐺𝑊))
2614bnj1424 31426 . . . 4 (𝑧𝐸 → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
2726adantl 474 . . 3 ((𝜒𝑧𝐸) → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
2819, 25, 27mpjaodan 982 . 2 ((𝜒𝑧𝐸) → (𝑄𝑧) = (𝐺𝑊))
2928ralrimiva 3147 1 (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wo 874  w3a 1108   = wceq 1653  wex 1875  wcel 2157  {cab 2785  wne 2971  wral 3089  wrex 3090  {crab 3093  [wsbc 3633  cun 3767  wss 3769  c0 4115  {csn 4368  cop 4374   cuni 4628   class class class wbr 4843  dom cdm 5312  cres 5314   Fn wfn 6096  cfv 6101  w-bnj17 31272   predc-bnj14 31274   FrSe w-bnj15 31278   trClc-bnj18 31280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-reg 8739  ax-inf2 8788
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7300  df-1o 7799  df-bnj17 31273  df-bnj14 31275  df-bnj13 31277  df-bnj15 31279  df-bnj18 31281
This theorem is referenced by:  bnj1312  31643
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