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Theorem bnj1442 32218
Description: Technical lemma for bnj60 32231. 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
bnj1442.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1442.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1442.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1442.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1442.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1442.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1442.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1442.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1442.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1442.10 𝑃 = 𝐻
bnj1442.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1442.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1442.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1442.14 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1442.15 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1442.16 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1442.17 (𝜃 ↔ (𝜒𝑧𝐸))
bnj1442.18 (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))
Assertion
Ref Expression
bnj1442 (𝜂 → (𝑄𝑧) = (𝐺𝑊))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜂(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑦,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑅(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐸(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1442
StepHypRef Expression
1 bnj1442.18 . . 3 (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))
2 bnj1442.17 . . . 4 (𝜃 ↔ (𝜒𝑧𝐸))
3 bnj1442.16 . . . . . 6 (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
43bnj930 31940 . . . . 5 (𝜒 → Fun 𝑄)
5 opex 5347 . . . . . . . 8 𝑥, (𝐺𝑍)⟩ ∈ V
65snid 4591 . . . . . . 7 𝑥, (𝐺𝑍)⟩ ∈ {⟨𝑥, (𝐺𝑍)⟩}
7 elun2 4150 . . . . . . 7 (⟨𝑥, (𝐺𝑍)⟩ ∈ {⟨𝑥, (𝐺𝑍)⟩} → ⟨𝑥, (𝐺𝑍)⟩ ∈ (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
86, 7ax-mp 5 . . . . . 6 𝑥, (𝐺𝑍)⟩ ∈ (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
9 bnj1442.12 . . . . . 6 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
108, 9eleqtrri 2909 . . . . 5 𝑥, (𝐺𝑍)⟩ ∈ 𝑄
11 funopfv 6710 . . . . 5 (Fun 𝑄 → (⟨𝑥, (𝐺𝑍)⟩ ∈ 𝑄 → (𝑄𝑥) = (𝐺𝑍)))
124, 10, 11mpisyl 21 . . . 4 (𝜒 → (𝑄𝑥) = (𝐺𝑍))
132, 12bnj832 31928 . . 3 (𝜃 → (𝑄𝑥) = (𝐺𝑍))
141, 13bnj832 31928 . 2 (𝜂 → (𝑄𝑥) = (𝐺𝑍))
15 elsni 4574 . . . 4 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
161, 15simplbiim 505 . . 3 (𝜂𝑧 = 𝑥)
1716fveq2d 6667 . 2 (𝜂 → (𝑄𝑧) = (𝑄𝑥))
18 bnj602 32086 . . . . . . . 8 (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
1918reseq2d 5846 . . . . . . 7 (𝑧 = 𝑥 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)))
2016, 19syl 17 . . . . . 6 (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)))
219bnj931 31941 . . . . . . . . . 10 𝑃𝑄
2221a1i 11 . . . . . . . . 9 (𝜒𝑃𝑄)
23 bnj1442.7 . . . . . . . . . . . 12 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
24 bnj1442.6 . . . . . . . . . . . . 13 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
2524simplbi 498 . . . . . . . . . . . 12 (𝜓𝑅 FrSe 𝐴)
2623, 25bnj835 31929 . . . . . . . . . . 11 (𝜒𝑅 FrSe 𝐴)
27 bnj1442.5 . . . . . . . . . . . 12 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2827, 23bnj1212 31970 . . . . . . . . . . 11 (𝜒𝑥𝐴)
29 bnj906 32101 . . . . . . . . . . 11 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3026, 28, 29syl2anc 584 . . . . . . . . . 10 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31 bnj1442.15 . . . . . . . . . . 11 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
32 fndm 6448 . . . . . . . . . . 11 (𝑃 Fn trCl(𝑥, 𝐴, 𝑅) → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
3331, 32syl 17 . . . . . . . . . 10 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
3430, 33sseqtrrd 4005 . . . . . . . . 9 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃)
354, 22, 34bnj1503 32020 . . . . . . . 8 (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
362, 35bnj832 31928 . . . . . . 7 (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
371, 36bnj832 31928 . . . . . 6 (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3820, 37eqtrd 2853 . . . . 5 (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3916, 38opeq12d 4803 . . . 4 (𝜂 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
40 bnj1442.13 . . . 4 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
41 bnj1442.11 . . . 4 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4239, 40, 413eqtr4g 2878 . . 3 (𝜂𝑊 = 𝑍)
4342fveq2d 6667 . 2 (𝜂 → (𝐺𝑊) = (𝐺𝑍))
4414, 17, 433eqtr4d 2863 1 (𝜂 → (𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  {crab 3139  [wsbc 3769  cun 3931  wss 3933  c0 4288  {csn 4557  cop 4563   cuni 4830   class class class wbr 5057  dom cdm 5548  cres 5550  Fun wfun 6342   Fn wfn 6343  cfv 6348   predc-bnj14 31857   FrSe w-bnj15 31861   trClc-bnj18 31863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1o 8091  df-bnj17 31856  df-bnj14 31858  df-bnj13 31860  df-bnj15 31862  df-bnj18 31864
This theorem is referenced by:  bnj1423  32220
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