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Theorem bnj1384 32292
Description: Technical lemma for bnj60 32322. 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 32290 . . . . 5 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1371 32289 . . . 4 (𝑓𝐻 → Fun 𝑓)
1312rgen 3146 . . 3 𝑓𝐻 Fun 𝑓
14 id 22 . . . . . 6 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
151, 2, 3, 4, 5, 6, 7, 8, 9bnj1374 32291 . . . . . 6 (𝑓𝐻𝑓𝐶)
16 nfab1 2977 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
179, 16nfcxfr 2973 . . . . . . . . 9 𝑓𝐻
1817nfcri 2969 . . . . . . . 8 𝑓 𝑔𝐻
19 nfab1 2977 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
203, 19nfcxfr 2973 . . . . . . . . 9 𝑓𝐶
2120nfcri 2969 . . . . . . . 8 𝑓 𝑔𝐶
2218, 21nfim 1890 . . . . . . 7 𝑓(𝑔𝐻𝑔𝐶)
23 eleq1w 2893 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐻𝑔𝐻))
24 eleq1w 2893 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
2523, 24imbi12d 347 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓𝐻𝑓𝐶) ↔ (𝑔𝐻𝑔𝐶)))
2622, 25, 15chvarfv 2234 . . . . . 6 (𝑔𝐻𝑔𝐶)
27 eqid 2819 . . . . . . 7 (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
281, 2, 3, 27bnj1326 32286 . . . . . 6 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
2914, 15, 26, 28syl3an 1154 . . . . 5 ((𝑅 FrSe 𝐴𝑓𝐻𝑔𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
30293expib 1116 . . . 4 (𝑅 FrSe 𝐴 → ((𝑓𝐻𝑔𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
3130ralrimivv 3188 . . 3 (𝑅 FrSe 𝐴 → ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
32 biid 263 . . . 4 (∀𝑓𝐻 Fun 𝑓 ↔ ∀𝑓𝐻 Fun 𝑓)
33 biid 263 . . . 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 32081 . . . 4 (𝑧𝐻 → ∀𝑓 𝑧𝐻)
3532, 27, 33, 34bnj1386 32093 . . 3 ((∀𝑓𝐻 Fun 𝑓 ∧ ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) → Fun 𝐻)
3613, 31, 35sylancr 589 . 2 (𝑅 FrSe 𝐴 → Fun 𝐻)
3710funeqi 6369 . 2 (Fun 𝑃 ↔ Fun 𝐻)
3836, 37sylibr 236 1 (𝑅 FrSe 𝐴 → Fun 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wex 1773  wcel 2107  {cab 2797  wne 3014  wral 3136  wrex 3137  {crab 3140  [wsbc 3770  cun 3932  cin 3933  wss 3934  c0 4289  {csn 4559  cop 4565   cuni 4830   class class class wbr 5057  dom cdm 5548  cres 5550  Fun wfun 6342   Fn wfn 6343  cfv 6348   predc-bnj14 31946   FrSe w-bnj15 31950   trClc-bnj18 31952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-reg 9048  ax-inf2 9096
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-1o 8094  df-bnj17 31945  df-bnj14 31947  df-bnj13 31949  df-bnj15 31951  df-bnj18 31953  df-bnj19 31955
This theorem is referenced by:  bnj1312  32318
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