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Theorem bnj1384 31438
Description: Technical lemma for bnj60 31468. 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 31436 . . . . 5 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1371 31435 . . . 4 (𝑓𝐻 → Fun 𝑓)
1312rgen 3071 . . 3 𝑓𝐻 Fun 𝑓
14 id 22 . . . . . 6 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
151, 2, 3, 4, 5, 6, 7, 8, 9bnj1374 31437 . . . . . 6 (𝑓𝐻𝑓𝐶)
16 nfab1 2915 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
179, 16nfcxfr 2911 . . . . . . . . 9 𝑓𝐻
1817nfcri 2907 . . . . . . . 8 𝑓 𝑔𝐻
19 nfab1 2915 . . . . . . . . . 10 𝑓{𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
203, 19nfcxfr 2911 . . . . . . . . 9 𝑓𝐶
2120nfcri 2907 . . . . . . . 8 𝑓 𝑔𝐶
2218, 21nfim 1977 . . . . . . 7 𝑓(𝑔𝐻𝑔𝐶)
23 eleq1w 2833 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐻𝑔𝐻))
24 eleq1w 2833 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝐶𝑔𝐶))
2523, 24imbi12d 333 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓𝐻𝑓𝐶) ↔ (𝑔𝐻𝑔𝐶)))
2622, 25, 15chvar 2424 . . . . . 6 (𝑔𝐻𝑔𝐶)
27 eqid 2771 . . . . . . 7 (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
281, 2, 3, 27bnj1326 31432 . . . . . 6 ((𝑅 FrSe 𝐴𝑓𝐶𝑔𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
2914, 15, 26, 28syl3an 1163 . . . . 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 3119 . . 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 31230 . . . 4 (𝑧𝐻 → ∀𝑓 𝑧𝐻)
3532, 27, 33, 34bnj1386 31242 . . 3 ((∀𝑓𝐻 Fun 𝑓 ∧ ∀𝑓𝐻𝑔𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) → Fun 𝐻)
3613, 31, 35sylancr 567 . 2 (𝑅 FrSe 𝐴 → Fun 𝐻)
3710funeqi 6052 . 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 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  {crab 3065  [wsbc 3587  cun 3721  cin 3722  wss 3723  c0 4063  {csn 4316  cop 4322   cuni 4574   class class class wbr 4786  dom cdm 5249  cres 5251  Fun wfun 6025   Fn wfn 6026  cfv 6031   predc-bnj14 31094   FrSe w-bnj15 31098   trClc-bnj18 31100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-reg 8653  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-1o 7713  df-bnj17 31093  df-bnj14 31095  df-bnj13 31097  df-bnj15 31099  df-bnj18 31101  df-bnj19 31103
This theorem is referenced by:  bnj1312  31464
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