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Theorem bnj1442 30177
Description: Technical lemma for bnj60 30190. 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 29900 . . . . 5 (𝜒 → Fun 𝑄)
5 opex 4853 . . . . . . . 8 𝑥, (𝐺𝑍)⟩ ∈ V
65snid 4154 . . . . . . 7 𝑥, (𝐺𝑍)⟩ ∈ {⟨𝑥, (𝐺𝑍)⟩}
7 elun2 3742 . . . . . . 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 2686 . . . . 5 𝑥, (𝐺𝑍)⟩ ∈ 𝑄
11 funopfv 6130 . . . . 5 (Fun 𝑄 → (⟨𝑥, (𝐺𝑍)⟩ ∈ 𝑄 → (𝑄𝑥) = (𝐺𝑍)))
124, 10, 11mpisyl 21 . . . 4 (𝜒 → (𝑄𝑥) = (𝐺𝑍))
132, 12bnj832 29888 . . 3 (𝜃 → (𝑄𝑥) = (𝐺𝑍))
141, 13bnj832 29888 . 2 (𝜂 → (𝑄𝑥) = (𝐺𝑍))
15 elsni 4141 . . . 4 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
161, 15simplbiim 656 . . 3 (𝜂𝑧 = 𝑥)
1716fveq2d 6092 . 2 (𝜂 → (𝑄𝑧) = (𝑄𝑥))
18 bnj602 30045 . . . . . . . 8 (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
1918reseq2d 5304 . . . . . . 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 29901 . . . . . . . . . 10 𝑃𝑄
2221a1i 11 . . . . . . . . 9 (𝜒𝑃𝑄)
23 bnj1442.7 . . . . . . . . . . . 12 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
24 bnj1442.6 . . . . . . . . . . . . 13 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
2524simplbi 474 . . . . . . . . . . . 12 (𝜓𝑅 FrSe 𝐴)
2623, 25bnj835 29889 . . . . . . . . . . 11 (𝜒𝑅 FrSe 𝐴)
27 bnj1442.5 . . . . . . . . . . . 12 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2827, 23bnj1212 29930 . . . . . . . . . . 11 (𝜒𝑥𝐴)
29 bnj906 30060 . . . . . . . . . . 11 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3026, 28, 29syl2anc 690 . . . . . . . . . 10 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31 bnj1442.15 . . . . . . . . . . 11 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
32 fndm 5890 . . . . . . . . . . 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, 33sseqtr4d 3604 . . . . . . . . 9 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃)
354, 22, 34bnj1503 29979 . . . . . . . 8 (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
362, 35bnj832 29888 . . . . . . 7 (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
371, 36bnj832 29888 . . . . . 6 (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3820, 37eqtrd 2643 . . . . 5 (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3916, 38opeq12d 4342 . . . 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 2668 . . 3 (𝜂𝑊 = 𝑍)
4342fveq2d 6092 . 2 (𝜂 → (𝐺𝑊) = (𝐺𝑍))
4414, 17, 433eqtr4d 2653 1 (𝜂 → (𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wne 2779  wral 2895  wrex 2896  {crab 2899  [wsbc 3401  cun 3537  wss 3539  c0 3873  {csn 4124  cop 4130   cuni 4366   class class class wbr 4577  dom cdm 5028  cres 5030  Fun wfun 5784   Fn wfn 5785  cfv 5790   predc-bnj14 29813   FrSe w-bnj15 29817   trClc-bnj18 29819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-reg 8357  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-1o 7424  df-bnj17 29812  df-bnj14 29814  df-bnj13 29816  df-bnj15 29818  df-bnj18 29820
This theorem is referenced by:  bnj1423  30179
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