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Theorem bnj1501 35060
Description: Technical lemma for bnj1500 35061. 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 35055 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
87fndmd 6674 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
91, 8bnj832 34751 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
102, 9eleqtrrd 2842 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
116dmeqi 5918 . . . . . . . 8 dom 𝐹 = dom 𝐶
125bnj1317 34814 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1312bnj1400 34828 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1411, 13eqtri 2763 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1510, 14eleqtrdi 2849 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1615bnj1405 34829 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
17 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1816, 17bnj1209 34789 . . . 4 (𝜑 → ∃𝑓𝜓)
195bnj1436 34832 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2019bnj1299 34811 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
21 fndm 6672 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2220, 21bnj31 34712 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2317, 22bnj836 34753 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
24 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
253, 4, 5, 6, 1, 17bnj1518 35057 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2623, 24, 25bnj1521 34844 . . . . . 6 (𝜓 → ∃𝑑𝜒)
277fnfund 6670 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
281, 27bnj832 34751 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
2917, 28bnj835 34752 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
30 elssuni 4942 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3130, 6sseqtrrdi 4047 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3217, 31bnj836 34753 . . . . . . . . . 10 (𝜓𝑓𝐹)
3317simp3bi 1146 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3429, 32, 33bnj1502 34841 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
353, 4, 5bnj1514 35056 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3617, 35bnj836 34753 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3736, 33bnj1294 34810 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3834, 37eqtrd 2775 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
3924, 38bnj835 34752 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4024, 29bnj835 34752 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4124, 32bnj835 34752 . . . . . . . . . . 11 (𝜒𝑓𝐹)
423bnj1517 34843 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4324, 42bnj836 34753 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4424, 33bnj835 34752 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4524simp3bi 1146 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4644, 45eleqtrd 2841 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4743, 46bnj1294 34810 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4847, 45sseqtrrd 4037 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
4940, 41, 48bnj1503 34842 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5049opeq2d 4885 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5150, 4eqtr4di 2793 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5251fveq2d 6911 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5339, 52eqtr4d 2778 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5426, 53bnj593 34738 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
553, 4, 5, 6bnj1519 35058 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5654, 55bnj1397 34827 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5718, 56bnj593 34738 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
583, 4, 5, 6bnj1520 35059 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5957, 58bnj1397 34827 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
601, 59bnj1459 34836 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 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963  cop 4637   cuni 4912   ciun 4996  dom cdm 5689  cres 5691  Fun wfun 6557   Fn wfn 6558  cfv 6563   predc-bnj14 34681   FrSe w-bnj15 34685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-bnj17 34680  df-bnj14 34682  df-bnj13 34684  df-bnj15 34686  df-bnj18 34688  df-bnj19 34690
This theorem is referenced by:  bnj1500  35061
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