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Theorem bnj1442 31449
Description: Technical lemma for bnj60 31462. 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 31172 . . . . 5 (𝜒 → Fun 𝑄)
5 opex 5060 . . . . . . . 8 𝑥, (𝐺𝑍)⟩ ∈ V
65snid 4345 . . . . . . 7 𝑥, (𝐺𝑍)⟩ ∈ {⟨𝑥, (𝐺𝑍)⟩}
7 elun2 3930 . . . . . . 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 2848 . . . . 5 𝑥, (𝐺𝑍)⟩ ∈ 𝑄
11 funopfv 6376 . . . . 5 (Fun 𝑄 → (⟨𝑥, (𝐺𝑍)⟩ ∈ 𝑄 → (𝑄𝑥) = (𝐺𝑍)))
124, 10, 11mpisyl 21 . . . 4 (𝜒 → (𝑄𝑥) = (𝐺𝑍))
132, 12bnj832 31160 . . 3 (𝜃 → (𝑄𝑥) = (𝐺𝑍))
141, 13bnj832 31160 . 2 (𝜂 → (𝑄𝑥) = (𝐺𝑍))
15 elsni 4331 . . . 4 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
161, 15simplbiim 487 . . 3 (𝜂𝑧 = 𝑥)
1716fveq2d 6336 . 2 (𝜂 → (𝑄𝑧) = (𝑄𝑥))
18 bnj602 31317 . . . . . . . 8 (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
1918reseq2d 5534 . . . . . . 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 31173 . . . . . . . . . 10 𝑃𝑄
2221a1i 11 . . . . . . . . 9 (𝜒𝑃𝑄)
23 bnj1442.7 . . . . . . . . . . . 12 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
24 bnj1442.6 . . . . . . . . . . . . 13 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
2524simplbi 479 . . . . . . . . . . . 12 (𝜓𝑅 FrSe 𝐴)
2623, 25bnj835 31161 . . . . . . . . . . 11 (𝜒𝑅 FrSe 𝐴)
27 bnj1442.5 . . . . . . . . . . . 12 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2827, 23bnj1212 31202 . . . . . . . . . . 11 (𝜒𝑥𝐴)
29 bnj906 31332 . . . . . . . . . . 11 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3026, 28, 29syl2anc 565 . . . . . . . . . 10 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31 bnj1442.15 . . . . . . . . . . 11 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
32 fndm 6130 . . . . . . . . . . 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 3789 . . . . . . . . 9 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃)
354, 22, 34bnj1503 31251 . . . . . . . 8 (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
362, 35bnj832 31160 . . . . . . 7 (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
371, 36bnj832 31160 . . . . . 6 (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3820, 37eqtrd 2804 . . . . 5 (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3916, 38opeq12d 4545 . . . 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 2829 . . 3 (𝜂𝑊 = 𝑍)
4342fveq2d 6336 . 2 (𝜂 → (𝐺𝑊) = (𝐺𝑍))
4414, 17, 433eqtr4d 2814 1 (𝜂 → (𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  {cab 2756  wne 2942  wral 3060  wrex 3061  {crab 3064  [wsbc 3585  cun 3719  wss 3721  c0 4061  {csn 4314  cop 4320   cuni 4572   class class class wbr 4784  dom cdm 5249  cres 5251  Fun wfun 6025   Fn wfn 6026  cfv 6031   predc-bnj14 31088   FrSe w-bnj15 31092   trClc-bnj18 31094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-reg 8652  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7212  df-1o 7712  df-bnj17 31087  df-bnj14 31089  df-bnj13 31091  df-bnj15 31093  df-bnj18 31095
This theorem is referenced by:  bnj1423  31451
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