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Theorem bnj1452 32552
 Description: Technical lemma for bnj60 32562. 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 32299 . . . . 5 (𝜒𝑥𝐴)
54snssd 4699 . . . 4 (𝜒 → {𝑥} ⊆ 𝐴)
6 bnj1147 32494 . . . . 5 trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴
76a1i 11 . . . 4 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴)
85, 7unssd 4091 . . 3 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ⊆ 𝐴)
91, 8eqsstrid 3940 . 2 (𝜒𝐸𝐴)
10 elsni 4539 . . . . . . . 8 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
1110adantl 485 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → 𝑧 = 𝑥)
12 bnj602 32415 . . . . . . 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 501 . . . . . . . . 9 (𝜓𝑅 FrSe 𝐴)
163, 15bnj835 32258 . . . . . . . 8 (𝜒𝑅 FrSe 𝐴)
17 bnj906 32430 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1816, 4, 17syl2anc 587 . . . . . . 7 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1918ad2antrr 725 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
2013, 19eqsstrd 3930 . . . . 5 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
21 ssun4 4080 . . . . . 6 ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2221, 1sseqtrrdi 3943 . . . . 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 725 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
25 simpr 488 . . . . . . . 8 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
266, 25bnj1213 32298 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧𝐴)
27 bnj906 32430 . . . . . . 7 ((𝑅 FrSe 𝐴𝑧𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
2824, 26, 27syl2anc 587 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
294ad2antrr 725 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑥𝐴)
30 bnj1125 32492 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3124, 29, 25, 30syl3anc 1368 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3228, 31sstrd 3902 . . . . 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 32338 . . . . 5 (𝑧𝐸 → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3534adantl 485 . . . 4 ((𝜒𝑧𝐸) → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3623, 33, 35mpjaodan 956 . . 3 ((𝜒𝑧𝐸) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
3736ralrimiva 3113 . 2 (𝜒 → ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
38 snex 5300 . . . . . . . 8 {𝑥} ∈ V
3938a1i 11 . . . . . . 7 (𝜒 → {𝑥} ∈ V)
40 bnj893 32428 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4116, 4, 40syl2anc 587 . . . . . . 7 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4239, 41bnj1149 32292 . . . . . 6 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ∈ V)
431, 42eqeltrid 2856 . . . . 5 (𝜒𝐸 ∈ V)
44 bnj1452.1 . . . . . 6 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4544bnj1454 32342 . . . . 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 32415 . . . . . . . 8 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
4847sseq1d 3923 . . . . . . 7 (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
4948cbvralvw 3361 . . . . . 6 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
5049anbi2i 625 . . . . 5 ((𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5150sbcbii 3753 . . . 4 ([𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ [𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5246, 51bitrdi 290 . . 3 (𝜒 → (𝐸𝐵[𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)))
53 sseq1 3917 . . . . . 6 (𝑑 = 𝐸 → (𝑑𝐴𝐸𝐴))
54 sseq2 3918 . . . . . . 7 (𝑑 = 𝐸 → ( pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5554raleqbi1dv 3321 . . . . . 6 (𝑑 = 𝐸 → (∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5653, 55anbi12d 633 . . . . 5 (𝑑 = 𝐸 → ((𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
5756sbcieg 3735 . . . 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 282 . 2 (𝜒 → (𝐸𝐵 ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
609, 37, 59mpbir2and 712 1 (𝜒𝐸𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409  [wsbc 3696   ∪ cun 3856   ⊆ wss 3858  ∅c0 4225  {csn 4522  ⟨cop 4528  ∪ cuni 4798   class class class wbr 5032  dom cdm 5524   ↾ cres 5526   Fn wfn 6330  ‘cfv 6335   predc-bnj14 32186   FrSe w-bnj15 32190   trClc-bnj18 32192 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-reg 9089  ax-inf2 9137 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 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-1o 8112  df-bnj17 32185  df-bnj14 32187  df-bnj13 32189  df-bnj15 32191  df-bnj18 32193  df-bnj19 32195 This theorem is referenced by:  bnj1312  32558
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