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Theorem bnj1421 30845
Description: Technical lemma for bnj60 30865. 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
bnj1421.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1421.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1421.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1421.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1421.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1421.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1421.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1421.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1421.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1421.10 𝑃 = 𝐻
bnj1421.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1421.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1421.13 (𝜒 → Fun 𝑃)
bnj1421.14 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1421.15 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
Assertion
Ref Expression
bnj1421 (𝜒 → Fun 𝑄)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑓,𝑑)   𝐴(𝑦,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑦,𝑓,𝑑)   𝐺(𝑥,𝑦,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1421
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj1421.13 . . . 4 (𝜒 → Fun 𝑃)
2 vex 3192 . . . . 5 𝑥 ∈ V
3 fvex 6163 . . . . 5 (𝐺𝑍) ∈ V
42, 3funsn 5902 . . . 4 Fun {⟨𝑥, (𝐺𝑍)⟩}
51, 4jctir 560 . . 3 (𝜒 → (Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺𝑍)⟩}))
6 bnj1421.15 . . . . 5 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
73dmsnop 5573 . . . . . 6 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
87a1i 11 . . . . 5 (𝜒 → dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥})
96, 8ineq12d 3798 . . . 4 (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}))
10 bnj1421.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
11 bnj1421.6 . . . . . . . 8 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
1211simplbi 476 . . . . . . 7 (𝜓𝑅 FrSe 𝐴)
1310, 12bnj835 30564 . . . . . 6 (𝜒𝑅 FrSe 𝐴)
14 biid 251 . . . . . . . 8 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
15 biid 251 . . . . . . . 8 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
16 biid 251 . . . . . . . 8 (∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)))
17 biid 251 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴 ∧ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑅 FrSe 𝐴𝑥𝐴 ∧ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))))
18 eqid 2621 . . . . . . . 8 ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑧 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑧, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑧 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑧, 𝐴, 𝑅))
1914, 15, 16, 17, 18bnj1417 30844 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
20 disjsn 4221 . . . . . . . 8 (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2120ralbii 2975 . . . . . . 7 (∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2219, 21sylibr 224 . . . . . 6 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
2313, 22syl 17 . . . . 5 (𝜒 → ∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
24 bnj1421.5 . . . . . 6 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2524, 10bnj1212 30605 . . . . 5 (𝜒𝑥𝐴)
2623, 25bnj1294 30623 . . . 4 (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
279, 26eqtrd 2655 . . 3 (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ∅)
28 funun 5895 . . 3 (((Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺𝑍)⟩}) ∧ (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ∅) → Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
295, 27, 28syl2anc 692 . 2 (𝜒 → Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
30 bnj1421.12 . . 3 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3130funeqi 5873 . 2 (Fun 𝑄 ↔ Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
3229, 31sylibr 224 1 (𝜒 → Fun 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  [wsbc 3421  cun 3557  cin 3558  wss 3559  c0 3896  {csn 4153  cop 4159   cuni 4407   ciun 4490   class class class wbr 4618  dom cdm 5079  cres 5081  Fun wfun 5846   Fn wfn 5847  cfv 5852   predc-bnj14 30488   FrSe w-bnj15 30492   trClc-bnj18 30494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8448  ax-inf2 8489
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-1o 7512  df-bnj17 30487  df-bnj14 30489  df-bnj13 30491  df-bnj15 30493  df-bnj18 30495  df-bnj19 30497
This theorem is referenced by:  bnj1312  30861
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