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Theorem bnj1384 30843
Description: Technical lemma for bnj60 30873. 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 30841 . . . . 5 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1371 30840 . . . 4 (𝑓𝐻 → Fun 𝑓)
1312rgen 2917 . . 3 𝑓𝐻 Fun 𝑓
14 id 22 . . . . . 6 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
151, 2, 3, 4, 5, 6, 7, 8, 9bnj1374 30842 . . . . . 6 (𝑓𝐻𝑓𝐶)
16 nfab1 2763 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
179, 16nfcxfr 2759 . . . . . . . . 9 𝑓𝐻
1817nfcri 2755 . . . . . . . 8 𝑓 𝑔𝐻
19 nfab1 2763 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
203, 19nfcxfr 2759 . . . . . . . . 9 𝑓𝐶
2120nfcri 2755 . . . . . . . 8 𝑓 𝑔𝐶
2218, 21nfim 1822 . . . . . . 7 𝑓(𝑔𝐻𝑔𝐶)
23 eleq1 2686 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐻𝑔𝐻))
24 eleq1 2686 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
2523, 24imbi12d 334 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓𝐻𝑓𝐶) ↔ (𝑔𝐻𝑔𝐶)))
2622, 25, 15chvar 2261 . . . . . 6 (𝑔𝐻𝑔𝐶)
27 eqid 2621 . . . . . . 7 (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
281, 2, 3, 27bnj1326 30837 . . . . . 6 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
2914, 15, 26, 28syl3an 1365 . . . . 5 ((𝑅 FrSe 𝐴𝑓𝐻𝑔𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
30293expib 1265 . . . 4 (𝑅 FrSe 𝐴 → ((𝑓𝐻𝑔𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
3130ralrimivv 2965 . . 3 (𝑅 FrSe 𝐴 → ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
32 biid 251 . . . 4 (∀𝑓𝐻 Fun 𝑓 ↔ ∀𝑓𝐻 Fun 𝑓)
33 biid 251 . . . 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 30635 . . . 4 (𝑧𝐻 → ∀𝑓 𝑧𝐻)
3532, 27, 33, 34bnj1386 30647 . . 3 ((∀𝑓𝐻 Fun 𝑓 ∧ ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) → Fun 𝐻)
3613, 31, 35sylancr 694 . 2 (𝑅 FrSe 𝐴 → Fun 𝐻)
3710funeqi 5873 . 2 (Fun 𝑃 ↔ Fun 𝐻)
3836, 37sylibr 224 1 (𝑅 FrSe 𝐴 → Fun 𝑃)
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   class class class wbr 4618  dom cdm 5079  cres 5081  Fun wfun 5846   Fn wfn 5847  cfv 5852   predc-bnj14 30496   FrSe w-bnj15 30500   trClc-bnj18 30502
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 8449  ax-inf2 8490
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 30495  df-bnj14 30497  df-bnj13 30499  df-bnj15 30501  df-bnj18 30503  df-bnj19 30505
This theorem is referenced by:  bnj1312  30869
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