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Theorem bnj1384 31226
Description: Technical lemma for bnj60 31256. 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 31224 . . . . 5 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1371 31223 . . . 4 (𝑓𝐻 → Fun 𝑓)
1312rgen 2951 . . 3 𝑓𝐻 Fun 𝑓
14 id 22 . . . . . 6 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
151, 2, 3, 4, 5, 6, 7, 8, 9bnj1374 31225 . . . . . 6 (𝑓𝐻𝑓𝐶)
16 nfab1 2795 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
179, 16nfcxfr 2791 . . . . . . . . 9 𝑓𝐻
1817nfcri 2787 . . . . . . . 8 𝑓 𝑔𝐻
19 nfab1 2795 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
203, 19nfcxfr 2791 . . . . . . . . 9 𝑓𝐶
2120nfcri 2787 . . . . . . . 8 𝑓 𝑔𝐶
2218, 21nfim 1865 . . . . . . 7 𝑓(𝑔𝐻𝑔𝐶)
23 eleq1 2718 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐻𝑔𝐻))
24 eleq1 2718 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
2523, 24imbi12d 333 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓𝐻𝑓𝐶) ↔ (𝑔𝐻𝑔𝐶)))
2622, 25, 15chvar 2298 . . . . . 6 (𝑔𝐻𝑔𝐶)
27 eqid 2651 . . . . . . 7 (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
281, 2, 3, 27bnj1326 31220 . . . . . 6 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
2914, 15, 26, 28syl3an 1408 . . . . 5 ((𝑅 FrSe 𝐴𝑓𝐻𝑔𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
30293expib 1287 . . . 4 (𝑅 FrSe 𝐴 → ((𝑓𝐻𝑔𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
3130ralrimivv 2999 . . 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 31018 . . . 4 (𝑧𝐻 → ∀𝑓 𝑧𝐻)
3532, 27, 33, 34bnj1386 31030 . . 3 ((∀𝑓𝐻 Fun 𝑓 ∧ ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) → Fun 𝐻)
3613, 31, 35sylancr 696 . 2 (𝑅 FrSe 𝐴 → Fun 𝐻)
3710funeqi 5947 . 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 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  [wsbc 3468  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685  dom cdm 5143  cres 5145  Fun wfun 5920   Fn wfn 5921  cfv 5926   predc-bnj14 30882   FrSe w-bnj15 30886   trClc-bnj18 30888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-bnj17 30881  df-bnj14 30883  df-bnj13 30885  df-bnj15 30887  df-bnj18 30889  df-bnj19 30891
This theorem is referenced by:  bnj1312  31252
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