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Theorem bnj1501 32223
Description: Technical lemma for bnj1500 32224. 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
bnj1501.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1501.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1501.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1501.4 𝐹 = 𝐶
bnj1501.5 (𝜑 ↔ (𝑅 FrSe 𝐴𝑥𝐴))
bnj1501.6 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
bnj1501.7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
Assertion
Ref Expression
bnj1501 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑅,𝑑,𝑓,𝑥   𝑌,𝑑   𝜑,𝑑,𝑓
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑓,𝑑)   𝜒(𝑥,𝑓,𝑑)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓)

Proof of Theorem bnj1501
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1501.5 . 2 (𝜑 ↔ (𝑅 FrSe 𝐴𝑥𝐴))
21simprbi 497 . . . . . . . 8 (𝜑𝑥𝐴)
3 bnj1501.1 . . . . . . . . . . 11 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4 bnj1501.2 . . . . . . . . . . 11 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
5 bnj1501.3 . . . . . . . . . . 11 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
6 bnj1501.4 . . . . . . . . . . 11 𝐹 = 𝐶
73, 4, 5, 6bnj60 32218 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
8 fndm 6452 . . . . . . . . . 10 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
97, 8syl 17 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
101, 9bnj832 31915 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
112, 10eleqtrrd 2921 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
126dmeqi 5772 . . . . . . . 8 dom 𝐹 = dom 𝐶
135bnj1317 31979 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1413bnj1400 31993 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1512, 14eqtri 2849 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1611, 15syl6eleq 2928 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1716bnj1405 31994 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
18 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1917, 18bnj1209 31954 . . . 4 (𝜑 → ∃𝑓𝜓)
205bnj1436 31997 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2120bnj1299 31976 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
22 fndm 6452 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2321, 22bnj31 31875 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2418, 23bnj836 31917 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
25 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
263, 4, 5, 6, 1, 18bnj1518 32220 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2724, 25, 26bnj1521 32009 . . . . . 6 (𝜓 → ∃𝑑𝜒)
287bnj930 31927 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
291, 28bnj832 31915 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
3018, 29bnj835 31916 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
31 elssuni 4866 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3231, 6sseqtrrdi 4022 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3318, 32bnj836 31917 . . . . . . . . . 10 (𝜓𝑓𝐹)
3418simp3bi 1141 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3530, 33, 34bnj1502 32006 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
363, 4, 5bnj1514 32219 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3718, 36bnj836 31917 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3837, 34bnj1294 31975 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3935, 38eqtrd 2861 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
4025, 39bnj835 31916 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4125, 30bnj835 31916 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4225, 33bnj835 31916 . . . . . . . . . . 11 (𝜒𝑓𝐹)
433bnj1517 32008 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4425, 43bnj836 31917 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4525, 34bnj835 31916 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4625simp3bi 1141 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4745, 46eleqtrd 2920 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4844, 47bnj1294 31975 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4948, 46sseqtrrd 4012 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
5041, 42, 49bnj1503 32007 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5150opeq2d 4809 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5251, 4syl6eqr 2879 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5352fveq2d 6671 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5440, 53eqtr4d 2864 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5527, 54bnj593 31902 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
563, 4, 5, 6bnj1519 32221 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5755, 56bnj1397 31992 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5819, 57bnj593 31902 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
593, 4, 5, 6bnj1520 32222 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
6058, 59bnj1397 31992 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
611, 60bnj1459 32001 1 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  {cab 2804  wral 3143  wrex 3144  wss 3940  cop 4570   cuni 4837   ciun 4917  dom cdm 5554  cres 5556  Fun wfun 6346   Fn wfn 6347  cfv 6352   predc-bnj14 31844   FrSe w-bnj15 31848
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-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-reg 9045  ax-inf2 9093
This theorem depends on definitions:  df-bi 208  df-an 397  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 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-1o 8093  df-bnj17 31843  df-bnj14 31845  df-bnj13 31847  df-bnj15 31849  df-bnj18 31851  df-bnj19 31853
This theorem is referenced by:  bnj1500  32224
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