Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1501 Structured version   Visualization version   GIF version

Theorem bnj1501 34078
Description: Technical lemma for bnj1500 34079. 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 498 . . . . . . . 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 34073 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝐹 Fn 𝐴)
87fndmd 6655 . . . . . . . . 9 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
91, 8bnj832 33769 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐴)
102, 9eleqtrrd 2837 . . . . . . 7 (𝜑𝑥 ∈ dom 𝐹)
116dmeqi 5905 . . . . . . . 8 dom 𝐹 = dom 𝐶
125bnj1317 33832 . . . . . . . . 9 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
1312bnj1400 33846 . . . . . . . 8 dom 𝐶 = 𝑓𝐶 dom 𝑓
1411, 13eqtri 2761 . . . . . . 7 dom 𝐹 = 𝑓𝐶 dom 𝑓
1510, 14eleqtrdi 2844 . . . . . 6 (𝜑𝑥 𝑓𝐶 dom 𝑓)
1615bnj1405 33847 . . . . 5 (𝜑 → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
17 bnj1501.6 . . . . 5 (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))
1816, 17bnj1209 33807 . . . 4 (𝜑 → ∃𝑓𝜓)
195bnj1436 33850 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
2019bnj1299 33829 . . . . . . . . 9 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
21 fndm 6653 . . . . . . . . 9 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
2220, 21bnj31 33730 . . . . . . . 8 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
2317, 22bnj836 33771 . . . . . . 7 (𝜓 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
24 bnj1501.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))
253, 4, 5, 6, 1, 17bnj1518 34075 . . . . . . 7 (𝜓 → ∀𝑑𝜓)
2623, 24, 25bnj1521 33862 . . . . . 6 (𝜓 → ∃𝑑𝜒)
277fnfund 6651 . . . . . . . . . . . 12 (𝑅 FrSe 𝐴 → Fun 𝐹)
281, 27bnj832 33769 . . . . . . . . . . 11 (𝜑 → Fun 𝐹)
2917, 28bnj835 33770 . . . . . . . . . 10 (𝜓 → Fun 𝐹)
30 elssuni 4942 . . . . . . . . . . . 12 (𝑓𝐶𝑓 𝐶)
3130, 6sseqtrrdi 4034 . . . . . . . . . . 11 (𝑓𝐶𝑓𝐹)
3217, 31bnj836 33771 . . . . . . . . . 10 (𝜓𝑓𝐹)
3317simp3bi 1148 . . . . . . . . . 10 (𝜓𝑥 ∈ dom 𝑓)
3429, 32, 33bnj1502 33859 . . . . . . . . 9 (𝜓 → (𝐹𝑥) = (𝑓𝑥))
353, 4, 5bnj1514 34074 . . . . . . . . . . 11 (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3617, 35bnj836 33771 . . . . . . . . . 10 (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))
3736, 33bnj1294 33828 . . . . . . . . 9 (𝜓 → (𝑓𝑥) = (𝐺𝑌))
3834, 37eqtrd 2773 . . . . . . . 8 (𝜓 → (𝐹𝑥) = (𝐺𝑌))
3924, 38bnj835 33770 . . . . . . 7 (𝜒 → (𝐹𝑥) = (𝐺𝑌))
4024, 29bnj835 33770 . . . . . . . . . . 11 (𝜒 → Fun 𝐹)
4124, 32bnj835 33770 . . . . . . . . . . 11 (𝜒𝑓𝐹)
423bnj1517 33861 . . . . . . . . . . . . . 14 (𝑑𝐵 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4324, 42bnj836 33771 . . . . . . . . . . . . 13 (𝜒 → ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4424, 33bnj835 33770 . . . . . . . . . . . . . 14 (𝜒𝑥 ∈ dom 𝑓)
4524simp3bi 1148 . . . . . . . . . . . . . 14 (𝜒 → dom 𝑓 = 𝑑)
4644, 45eleqtrd 2836 . . . . . . . . . . . . 13 (𝜒𝑥𝑑)
4743, 46bnj1294 33828 . . . . . . . . . . . 12 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
4847, 45sseqtrrd 4024 . . . . . . . . . . 11 (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
4940, 41, 48bnj1503 33860 . . . . . . . . . 10 (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5049opeq2d 4881 . . . . . . . . 9 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
5150, 4eqtr4di 2791 . . . . . . . 8 (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
5251fveq2d 6896 . . . . . . 7 (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺𝑌))
5339, 52eqtr4d 2776 . . . . . 6 (𝜒 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5426, 53bnj593 33756 . . . . 5 (𝜓 → ∃𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
553, 4, 5, 6bnj1519 34076 . . . . 5 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5654, 55bnj1397 33845 . . . 4 (𝜓 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5718, 56bnj593 33756 . . 3 (𝜑 → ∃𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
583, 4, 5, 6bnj1520 34077 . . 3 ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
5957, 58bnj1397 33845 . 2 (𝜑 → (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
601, 59bnj1459 33854 1 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wrex 3071  wss 3949  cop 4635   cuni 4909   ciun 4998  dom cdm 5677  cres 5679  Fun wfun 6538   Fn wfn 6539  cfv 6544   predc-bnj14 33699   FrSe w-bnj15 33703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-bnj17 33698  df-bnj14 33700  df-bnj13 33702  df-bnj15 33704  df-bnj18 33706  df-bnj19 33708
This theorem is referenced by:  bnj1500  34079
  Copyright terms: Public domain W3C validator