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Theorem bnj1501 35081
Description: Technical lemma for bnj1500 35082. 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 496 . . . . . . . 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 35076 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
87fndmd 6673 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
91, 8bnj832 34772 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
102, 9eleqtrrd 2844 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
116dmeqi 5915 . . . . . . . 8 dom 𝐹 = dom 𝐶
125bnj1317 34835 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1312bnj1400 34849 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1411, 13eqtri 2765 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1510, 14eleqtrdi 2851 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1615bnj1405 34850 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
17 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1816, 17bnj1209 34810 . . . 4 (𝜑 → ∃𝑓𝜓)
195bnj1436 34853 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2019bnj1299 34832 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
21 fndm 6671 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2220, 21bnj31 34733 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2317, 22bnj836 34774 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
24 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
253, 4, 5, 6, 1, 17bnj1518 35078 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2623, 24, 25bnj1521 34865 . . . . . 6 (𝜓 → ∃𝑑𝜒)
277fnfund 6669 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
281, 27bnj832 34772 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
2917, 28bnj835 34773 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
30 elssuni 4937 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3130, 6sseqtrrdi 4025 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3217, 31bnj836 34774 . . . . . . . . . 10 (𝜓𝑓𝐹)
3317simp3bi 1148 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3429, 32, 33bnj1502 34862 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
353, 4, 5bnj1514 35077 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3617, 35bnj836 34774 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3736, 33bnj1294 34831 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3834, 37eqtrd 2777 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
3924, 38bnj835 34773 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4024, 29bnj835 34773 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4124, 32bnj835 34773 . . . . . . . . . . 11 (𝜒𝑓𝐹)
423bnj1517 34864 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4324, 42bnj836 34774 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4424, 33bnj835 34773 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4524simp3bi 1148 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4644, 45eleqtrd 2843 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4743, 46bnj1294 34831 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4847, 45sseqtrrd 4021 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
4940, 41, 48bnj1503 34863 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5049opeq2d 4880 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5150, 4eqtr4di 2795 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5251fveq2d 6910 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5339, 52eqtr4d 2780 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5426, 53bnj593 34759 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
553, 4, 5, 6bnj1519 35079 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5654, 55bnj1397 34848 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5718, 56bnj593 34759 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
583, 4, 5, 6bnj1520 35080 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5957, 58bnj1397 34848 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
601, 59bnj1459 34857 1 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  wss 3951  cop 4632   cuni 4907   ciun 4991  dom cdm 5685  cres 5687  Fun wfun 6555   Fn wfn 6556  cfv 6561   predc-bnj14 34702   FrSe w-bnj15 34706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-bnj17 34701  df-bnj14 34703  df-bnj13 34705  df-bnj15 34707  df-bnj18 34709  df-bnj19 34711
This theorem is referenced by:  bnj1500  35082
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