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Theorem bnj1452 30374
Description: Technical lemma for bnj60 30384. 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 30124 . . . . 5 (𝜒𝑥𝐴)
54snssd 4281 . . . 4 (𝜒 → {𝑥} ⊆ 𝐴)
6 bnj1147 30316 . . . . 5 trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴
76a1i 11 . . . 4 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴)
85, 7unssd 3751 . . 3 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ⊆ 𝐴)
91, 8syl5eqss 3612 . 2 (𝜒𝐸𝐴)
10 elsni 4142 . . . . . . . 8 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
1110adantl 481 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → 𝑧 = 𝑥)
12 bnj602 30239 . . . . . . 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 475 . . . . . . . . 9 (𝜓𝑅 FrSe 𝐴)
163, 15bnj835 30083 . . . . . . . 8 (𝜒𝑅 FrSe 𝐴)
17 bnj906 30254 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1816, 4, 17syl2anc 691 . . . . . . 7 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1918ad2antrr 758 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
2013, 19eqsstrd 3602 . . . . 5 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
21 ssun4 3741 . . . . . 6 ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2221, 1syl6sseqr 3615 . . . . 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 758 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
25 simpr 476 . . . . . . . 8 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
266, 25bnj1213 30123 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧𝐴)
27 bnj906 30254 . . . . . . 7 ((𝑅 FrSe 𝐴𝑧𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
2824, 26, 27syl2anc 691 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
294ad2antrr 758 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑥𝐴)
30 bnj1125 30314 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3124, 29, 25, 30syl3anc 1318 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3228, 31sstrd 3578 . . . . 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 30163 . . . . 5 (𝑧𝐸 → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3534adantl 481 . . . 4 ((𝜒𝑧𝐸) → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3623, 33, 35mpjaodan 823 . . 3 ((𝜒𝑧𝐸) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
3736ralrimiva 2949 . 2 (𝜒 → ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
38 snex 4835 . . . . . . . 8 {𝑥} ∈ V
3938a1i 11 . . . . . . 7 (𝜒 → {𝑥} ∈ V)
40 bnj893 30252 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4116, 4, 40syl2anc 691 . . . . . . 7 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4239, 41bnj1149 30117 . . . . . 6 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ∈ V)
431, 42syl5eqel 2692 . . . . 5 (𝜒𝐸 ∈ V)
44 bnj1452.1 . . . . . 6 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4544bnj1454 30166 . . . . 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 30239 . . . . . . . 8 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
4847sseq1d 3595 . . . . . . 7 (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
4948cbvralv 3147 . . . . . 6 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
5049anbi2i 726 . . . . 5 ((𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5150sbcbii 3458 . . . 4 ([𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ [𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5246, 51syl6bb 275 . . 3 (𝜒 → (𝐸𝐵[𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)))
53 sseq1 3589 . . . . . 6 (𝑑 = 𝐸 → (𝑑𝐴𝐸𝐴))
54 sseq2 3590 . . . . . . 7 (𝑑 = 𝐸 → ( pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5554raleqbi1dv 3123 . . . . . 6 (𝑑 = 𝐸 → (∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5653, 55anbi12d 743 . . . . 5 (𝑑 = 𝐸 → ((𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
5756sbcieg 3435 . . . 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 267 . 2 (𝜒 → (𝐸𝐵 ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
609, 37, 59mpbir2and 959 1 (𝜒𝐸𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wne 2780  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  [wsbc 3402  cun 3538  wss 3540  c0 3874  {csn 4125  cop 4131   cuni 4372   class class class wbr 4583  dom cdm 5038  cres 5040   Fn wfn 5799  cfv 5804   predc-bnj14 30007   FrSe w-bnj15 30011   trClc-bnj18 30013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016
This theorem is referenced by:  bnj1312  30380
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