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Theorem bnj1452 31500
Description: Technical lemma for bnj60 31510. 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
bnj1452.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1452.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1452.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1452.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1452.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1452.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1452.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1452.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1452.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1452.10 𝑃 = 𝐻
bnj1452.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1452.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1452.13 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1452.14 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
Assertion
Ref Expression
bnj1452 (𝜒𝐸𝐵)
Distinct variable groups:   𝐴,𝑑,𝑥,𝑧   𝐸,𝑑,𝑧   𝑅,𝑑,𝑥,𝑧   𝜒,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑦,𝑓)   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(𝑦,𝑓)   𝐸(𝑥,𝑦,𝑓)   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 bnj1452
StepHypRef Expression
1 bnj1452.14 . . 3 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
2 bnj1452.5 . . . . . 6 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
3 bnj1452.7 . . . . . 6 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
42, 3bnj1212 31250 . . . . 5 (𝜒𝑥𝐴)
54snssd 4494 . . . 4 (𝜒 → {𝑥} ⊆ 𝐴)
6 bnj1147 31442 . . . . 5 trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴
76a1i 11 . . . 4 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴)
85, 7unssd 3951 . . 3 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ⊆ 𝐴)
91, 8syl5eqss 3809 . 2 (𝜒𝐸𝐴)
10 elsni 4351 . . . . . . . 8 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
1110adantl 473 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → 𝑧 = 𝑥)
12 bnj602 31365 . . . . . . 7 (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
1311, 12syl 17 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
14 bnj1452.6 . . . . . . . . . 10 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
1514simplbi 491 . . . . . . . . 9 (𝜓𝑅 FrSe 𝐴)
163, 15bnj835 31209 . . . . . . . 8 (𝜒𝑅 FrSe 𝐴)
17 bnj906 31380 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1816, 4, 17syl2anc 579 . . . . . . 7 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1918ad2antrr 717 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
2013, 19eqsstrd 3799 . . . . 5 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
21 ssun4 3941 . . . . . 6 ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2221, 1syl6sseqr 3812 . . . . 5 ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
2320, 22syl 17 . . . 4 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
2416ad2antrr 717 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
25 simpr 477 . . . . . . . 8 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
266, 25bnj1213 31249 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧𝐴)
27 bnj906 31380 . . . . . . 7 ((𝑅 FrSe 𝐴𝑧𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
2824, 26, 27syl2anc 579 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
294ad2antrr 717 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑥𝐴)
30 bnj1125 31440 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3124, 29, 25, 30syl3anc 1490 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3228, 31sstrd 3771 . . . . 5 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3332, 22syl 17 . . . 4 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
341bnj1424 31289 . . . . 5 (𝑧𝐸 → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3534adantl 473 . . . 4 ((𝜒𝑧𝐸) → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3623, 33, 35mpjaodan 981 . . 3 ((𝜒𝑧𝐸) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
3736ralrimiva 3113 . 2 (𝜒 → ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
38 snex 5064 . . . . . . . 8 {𝑥} ∈ V
3938a1i 11 . . . . . . 7 (𝜒 → {𝑥} ∈ V)
40 bnj893 31378 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4116, 4, 40syl2anc 579 . . . . . . 7 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4239, 41bnj1149 31243 . . . . . 6 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ∈ V)
431, 42syl5eqel 2848 . . . . 5 (𝜒𝐸 ∈ V)
44 bnj1452.1 . . . . . 6 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4544bnj1454 31292 . . . . 5 (𝐸 ∈ V → (𝐸𝐵[𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)))
4643, 45syl 17 . . . 4 (𝜒 → (𝐸𝐵[𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)))
47 bnj602 31365 . . . . . . . 8 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
4847sseq1d 3792 . . . . . . 7 (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
4948cbvralv 3319 . . . . . 6 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
5049anbi2i 616 . . . . 5 ((𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5150sbcbii 3652 . . . 4 ([𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ [𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5246, 51syl6bb 278 . . 3 (𝜒 → (𝐸𝐵[𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)))
53 sseq1 3786 . . . . . 6 (𝑑 = 𝐸 → (𝑑𝐴𝐸𝐴))
54 sseq2 3787 . . . . . . 7 (𝑑 = 𝐸 → ( pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5554raleqbi1dv 3294 . . . . . 6 (𝑑 = 𝐸 → (∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5653, 55anbi12d 624 . . . . 5 (𝑑 = 𝐸 → ((𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
5756sbcieg 3629 . . . 4 (𝐸 ∈ V → ([𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
5843, 57syl 17 . . 3 (𝜒 → ([𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
5952, 58bitrd 270 . 2 (𝜒 → (𝐸𝐵 ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
609, 37, 59mpbir2and 704 1 (𝜒𝐸𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  [wsbc 3596  cun 3730  wss 3732  c0 4079  {csn 4334  cop 4340   cuni 4594   class class class wbr 4809  dom cdm 5277  cres 5279   Fn wfn 6063  cfv 6068   predc-bnj14 31137   FrSe w-bnj15 31141   trClc-bnj18 31143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-bnj17 31136  df-bnj14 31138  df-bnj13 31140  df-bnj15 31142  df-bnj18 31144  df-bnj19 31146
This theorem is referenced by:  bnj1312  31506
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