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Theorem bnj1442 31102
Description: Technical lemma for bnj60 31115. 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 30825 . . . . 5 (𝜒 → Fun 𝑄)
5 opex 4930 . . . . . . . 8 𝑥, (𝐺𝑍)⟩ ∈ V
65snid 4206 . . . . . . 7 𝑥, (𝐺𝑍)⟩ ∈ {⟨𝑥, (𝐺𝑍)⟩}
7 elun2 3779 . . . . . . 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 2699 . . . . 5 𝑥, (𝐺𝑍)⟩ ∈ 𝑄
11 funopfv 6233 . . . . 5 (Fun 𝑄 → (⟨𝑥, (𝐺𝑍)⟩ ∈ 𝑄 → (𝑄𝑥) = (𝐺𝑍)))
124, 10, 11mpisyl 21 . . . 4 (𝜒 → (𝑄𝑥) = (𝐺𝑍))
132, 12bnj832 30813 . . 3 (𝜃 → (𝑄𝑥) = (𝐺𝑍))
141, 13bnj832 30813 . 2 (𝜂 → (𝑄𝑥) = (𝐺𝑍))
15 elsni 4192 . . . 4 (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
161, 15simplbiim 659 . . 3 (𝜂𝑧 = 𝑥)
1716fveq2d 6193 . 2 (𝜂 → (𝑄𝑧) = (𝑄𝑥))
18 bnj602 30970 . . . . . . . 8 (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
1918reseq2d 5394 . . . . . . 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 30826 . . . . . . . . . 10 𝑃𝑄
2221a1i 11 . . . . . . . . 9 (𝜒𝑃𝑄)
23 bnj1442.7 . . . . . . . . . . . 12 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
24 bnj1442.6 . . . . . . . . . . . . 13 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
2524simplbi 476 . . . . . . . . . . . 12 (𝜓𝑅 FrSe 𝐴)
2623, 25bnj835 30814 . . . . . . . . . . 11 (𝜒𝑅 FrSe 𝐴)
27 bnj1442.5 . . . . . . . . . . . 12 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2827, 23bnj1212 30855 . . . . . . . . . . 11 (𝜒𝑥𝐴)
29 bnj906 30985 . . . . . . . . . . 11 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
3026, 28, 29syl2anc 693 . . . . . . . . . 10 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31 bnj1442.15 . . . . . . . . . . 11 (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
32 fndm 5988 . . . . . . . . . . 11 (𝑃 Fn trCl(𝑥, 𝐴, 𝑅) → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
3331, 32syl 17 . . . . . . . . . 10 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
3430, 33sseqtr4d 3640 . . . . . . . . 9 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃)
354, 22, 34bnj1503 30904 . . . . . . . 8 (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
362, 35bnj832 30813 . . . . . . 7 (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
371, 36bnj832 30813 . . . . . 6 (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3820, 37eqtrd 2655 . . . . 5 (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
3916, 38opeq12d 4408 . . . 4 (𝜂 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
40 bnj1442.13 . . . 4 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
41 bnj1442.11 . . . 4 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4239, 40, 413eqtr4g 2680 . . 3 (𝜂𝑊 = 𝑍)
4342fveq2d 6193 . 2 (𝜂 → (𝐺𝑊) = (𝐺𝑍))
4414, 17, 433eqtr4d 2665 1 (𝜂 → (𝑄𝑧) = (𝐺𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wex 1703  wcel 1989  {cab 2607  wne 2793  wral 2911  wrex 2912  {crab 2915  [wsbc 3433  cun 3570  wss 3572  c0 3913  {csn 4175  cop 4181   cuni 4434   class class class wbr 4651  dom cdm 5112  cres 5114  Fun wfun 5880   Fn wfn 5881  cfv 5886   predc-bnj14 30739   FrSe w-bnj15 30743   trClc-bnj18 30745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-reg 8494  ax-inf2 8535
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-om 7063  df-1o 7557  df-bnj17 30738  df-bnj14 30740  df-bnj13 30742  df-bnj15 30744  df-bnj18 30746
This theorem is referenced by:  bnj1423  31104
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