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Theorem bnj1452 31419
Description: Technical lemma for bnj60 31429. 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 31169 . . . . 5 (𝜒𝑥𝐴)
54snssd 4477 . . . 4 (𝜒 → {𝑥} ⊆ 𝐴)
6 bnj1147 31361 . . . . 5 trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴
76a1i 11 . . . 4 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴)
85, 7unssd 3924 . . 3 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ⊆ 𝐴)
91, 8syl5eqss 3782 . 2 (𝜒𝐸𝐴)
10 elsni 4330 . . . . . . . 8 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
1110adantl 473 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → 𝑧 = 𝑥)
12 bnj602 31284 . . . . . . 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 478 . . . . . . . . 9 (𝜓𝑅 FrSe 𝐴)
163, 15bnj835 31128 . . . . . . . 8 (𝜒𝑅 FrSe 𝐴)
17 bnj906 31299 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1816, 4, 17syl2anc 696 . . . . . . 7 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
1918ad2antrr 764 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
2013, 19eqsstrd 3772 . . . . 5 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
21 ssun4 3914 . . . . . 6 ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
2221, 1syl6sseqr 3785 . . . . 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 764 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
25 simpr 479 . . . . . . . 8 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
266, 25bnj1213 31168 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧𝐴)
27 bnj906 31299 . . . . . . 7 ((𝑅 FrSe 𝐴𝑧𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
2824, 26, 27syl2anc 696 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
294ad2antrr 764 . . . . . . 7 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑥𝐴)
30 bnj1125 31359 . . . . . . 7 ((𝑅 FrSe 𝐴𝑥𝐴𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3124, 29, 25, 30syl3anc 1473 . . . . . 6 (((𝜒𝑧𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3228, 31sstrd 3746 . . . . 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 31208 . . . . 5 (𝑧𝐸 → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3534adantl 473 . . . 4 ((𝜒𝑧𝐸) → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3623, 33, 35mpjaodan 862 . . 3 ((𝜒𝑧𝐸) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
3736ralrimiva 3096 . 2 (𝜒 → ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
38 snex 5049 . . . . . . . 8 {𝑥} ∈ V
3938a1i 11 . . . . . . 7 (𝜒 → {𝑥} ∈ V)
40 bnj893 31297 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4116, 4, 40syl2anc 696 . . . . . . 7 (𝜒 → trCl(𝑥, 𝐴, 𝑅) ∈ V)
4239, 41bnj1149 31162 . . . . . 6 (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ∈ V)
431, 42syl5eqel 2835 . . . . 5 (𝜒𝐸 ∈ V)
44 bnj1452.1 . . . . . 6 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4544bnj1454 31211 . . . . 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 31284 . . . . . . . 8 (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
4847sseq1d 3765 . . . . . . 7 (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
4948cbvralv 3302 . . . . . 6 (∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
5049anbi2i 732 . . . . 5 ((𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5150sbcbii 3624 . . . 4 ([𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ [𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
5246, 51syl6bb 276 . . 3 (𝜒 → (𝐸𝐵[𝐸 / 𝑑](𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)))
53 sseq1 3759 . . . . . 6 (𝑑 = 𝐸 → (𝑑𝐴𝐸𝐴))
54 sseq2 3760 . . . . . . 7 (𝑑 = 𝐸 → ( pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5554raleqbi1dv 3277 . . . . . 6 (𝑑 = 𝐸 → (∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
5653, 55anbi12d 749 . . . . 5 (𝑑 = 𝐸 → ((𝑑𝐴 ∧ ∀𝑧𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
5756sbcieg 3601 . . . 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 268 . 2 (𝜒 → (𝐸𝐵 ↔ (𝐸𝐴 ∧ ∀𝑧𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
609, 37, 59mpbir2and 995 1 (𝜒𝐸𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1624  wex 1845  wcel 2131  {cab 2738  wne 2924  wral 3042  wrex 3043  {crab 3046  Vcvv 3332  [wsbc 3568  cun 3705  wss 3707  c0 4050  {csn 4313  cop 4319   cuni 4580   class class class wbr 4796  dom cdm 5258  cres 5260   Fn wfn 6036  cfv 6041   predc-bnj14 31055   FrSe w-bnj15 31059   trClc-bnj18 31061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-reg 8654  ax-inf2 8703
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-1o 7721  df-bnj17 31054  df-bnj14 31056  df-bnj13 31058  df-bnj15 31060  df-bnj18 31062  df-bnj19 31064
This theorem is referenced by:  bnj1312  31425
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