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Theorem bnj1384 35025
Description: Technical lemma for bnj60 35055. 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
bnj1384.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1384.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1384.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1384.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1384.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1384.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1384.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1384.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1384.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1384.10 𝑃 = 𝐻
Assertion
Ref Expression
bnj1384 (𝑅 FrSe 𝐴 → Fun 𝑃)
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝑦,𝐶   𝐺,𝑑,𝑓   𝑅,𝑑,𝑓,𝑥   𝑦,𝑓,𝑥
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑓,𝑑)   𝐴(𝑦)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1384
Dummy variables 𝑧 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1384.1 . . . . 5 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2 bnj1384.2 . . . . 5 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3 bnj1384.3 . . . . 5 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
4 bnj1384.4 . . . . 5 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5 bnj1384.5 . . . . 5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
6 bnj1384.6 . . . . 5 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
7 bnj1384.7 . . . . 5 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
8 bnj1384.8 . . . . 5 (𝜏′[𝑦 / 𝑥]𝜏)
9 bnj1384.9 . . . . 5 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
10 bnj1384.10 . . . . 5 𝑃 = 𝐻
111, 2, 3, 4, 8bnj1373 35023 . . . . 5 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1371 35022 . . . 4 (𝑓𝐻 → Fun 𝑓)
1312rgen 3061 . . 3 𝑓𝐻 Fun 𝑓
14 id 22 . . . . . 6 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
151, 2, 3, 4, 5, 6, 7, 8, 9bnj1374 35024 . . . . . 6 (𝑓𝐻𝑓𝐶)
16 nfab1 2905 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
179, 16nfcxfr 2901 . . . . . . . . 9 𝑓𝐻
1817nfcri 2895 . . . . . . . 8 𝑓 𝑔𝐻
19 nfab1 2905 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
203, 19nfcxfr 2901 . . . . . . . . 9 𝑓𝐶
2120nfcri 2895 . . . . . . . 8 𝑓 𝑔𝐶
2218, 21nfim 1894 . . . . . . 7 𝑓(𝑔𝐻𝑔𝐶)
23 eleq1w 2822 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐻𝑔𝐻))
24 eleq1w 2822 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
2523, 24imbi12d 344 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓𝐻𝑓𝐶) ↔ (𝑔𝐻𝑔𝐶)))
2622, 25, 15chvarfv 2238 . . . . . 6 (𝑔𝐻𝑔𝐶)
27 eqid 2735 . . . . . . 7 (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
281, 2, 3, 27bnj1326 35019 . . . . . 6 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
2914, 15, 26, 28syl3an 1159 . . . . 5 ((𝑅 FrSe 𝐴𝑓𝐻𝑔𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
30293expib 1121 . . . 4 (𝑅 FrSe 𝐴 → ((𝑓𝐻𝑔𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
3130ralrimivv 3198 . . 3 (𝑅 FrSe 𝐴 → ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
32 biid 261 . . . 4 (∀𝑓𝐻 Fun 𝑓 ↔ ∀𝑓𝐻 Fun 𝑓)
33 biid 261 . . . 4 ((∀𝑓𝐻 Fun 𝑓 ∧ ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) ↔ (∀𝑓𝐻 Fun 𝑓 ∧ ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
349bnj1317 34814 . . . 4 (𝑧𝐻 → ∀𝑓 𝑧𝐻)
3532, 27, 33, 34bnj1386 34826 . . 3 ((∀𝑓𝐻 Fun 𝑓 ∧ ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) → Fun 𝐻)
3613, 31, 35sylancr 587 . 2 (𝑅 FrSe 𝐴 → Fun 𝐻)
3710funeqi 6589 . 2 (Fun 𝑃 ↔ Fun 𝐻)
3836, 37sylibr 234 1 (𝑅 FrSe 𝐴 → Fun 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  [wsbc 3791  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631  cop 4637   cuni 4912   class class class wbr 5148  dom cdm 5689  cres 5691  Fun wfun 6557   Fn wfn 6558  cfv 6563   predc-bnj14 34681   FrSe w-bnj15 34685   trClc-bnj18 34687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-bnj17 34680  df-bnj14 34682  df-bnj13 34684  df-bnj15 34686  df-bnj18 34688  df-bnj19 34690
This theorem is referenced by:  bnj1312  35051
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