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Theorem bnj1442 32435
Description: Technical lemma for bnj60 32448. 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 32155 . . . . 5 (𝜒 → Fun 𝑄)
5 opex 5324 . . . . . . . 8 𝑥, (𝐺𝑍)⟩ ∈ V
65snid 4564 . . . . . . 7 𝑥, (𝐺𝑍)⟩ ∈ {⟨𝑥, (𝐺𝑍)⟩}
7 elun2 4107 . . . . . . 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 2892 . . . . 5 𝑥, (𝐺𝑍)⟩ ∈ 𝑄
11 funopfv 6696 . . . . 5 (Fun 𝑄 → (⟨𝑥, (𝐺𝑍)⟩ ∈ 𝑄 → (𝑄𝑥) = (𝐺𝑍)))
124, 10, 11mpisyl 21 . . . 4 (𝜒 → (𝑄𝑥) = (𝐺𝑍))
132, 12bnj832 32143 . . 3 (𝜃 → (𝑄𝑥) = (𝐺𝑍))
141, 13bnj832 32143 . 2 (𝜂 → (𝑄𝑥) = (𝐺𝑍))
15 elsni 4545 . . . 4 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
161, 15simplbiim 508 . . 3 (𝜂𝑧 = 𝑥)
1716fveq2d 6653 . 2 (𝜂 → (𝑄𝑧) = (𝑄𝑥))
18 bnj602 32301 . . . . . . . 8 (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
1918reseq2d 5822 . . . . . . 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 32156 . . . . . . . . . 10 𝑃𝑄
2221a1i 11 . . . . . . . . 9 (𝜒𝑃𝑄)
23 bnj1442.7 . . . . . . . . . . . 12 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
24 bnj1442.6 . . . . . . . . . . . . 13 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
2524simplbi 501 . . . . . . . . . . . 12 (𝜓𝑅 FrSe 𝐴)
2623, 25bnj835 32144 . . . . . . . . . . 11 (𝜒𝑅 FrSe 𝐴)
27 bnj1442.5 . . . . . . . . . . . 12 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2827, 23bnj1212 32185 . . . . . . . . . . 11 (𝜒𝑥𝐴)
29 bnj906 32316 . . . . . . . . . . 11 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3026, 28, 29syl2anc 587 . . . . . . . . . 10 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31 bnj1442.15 . . . . . . . . . . 11 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
3231fndmd 6431 . . . . . . . . . 10 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
3330, 32sseqtrrd 3959 . . . . . . . . 9 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃)
344, 22, 33bnj1503 32235 . . . . . . . 8 (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
352, 34bnj832 32143 . . . . . . 7 (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
361, 35bnj832 32143 . . . . . 6 (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3720, 36eqtrd 2836 . . . . 5 (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3816, 37opeq12d 4776 . . . 4 (𝜂 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
39 bnj1442.13 . . . 4 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
40 bnj1442.11 . . . 4 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4138, 39, 403eqtr4g 2861 . . 3 (𝜂𝑊 = 𝑍)
4241fveq2d 6653 . 2 (𝜂 → (𝐺𝑊) = (𝐺𝑍))
4314, 17, 423eqtr4d 2846 1 (𝜂 → (𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wne 2990  wral 3109  wrex 3110  {crab 3113  [wsbc 3723  cun 3882  wss 3884  c0 4246  {csn 4528  cop 4534   cuni 4803   class class class wbr 5033  dom cdm 5523  cres 5525  Fun wfun 6322   Fn wfn 6323  cfv 6328   predc-bnj14 32072   FrSe w-bnj15 32076   trClc-bnj18 32078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-1o 8089  df-bnj17 32071  df-bnj14 32073  df-bnj13 32075  df-bnj15 32077  df-bnj18 32079
This theorem is referenced by:  bnj1423  32437
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