Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1296 Structured version   Visualization version   GIF version

Theorem bnj1296 31427
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
bnj1296.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1296.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1296.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1296.4 𝐷 = (dom 𝑔 ∩ dom )
bnj1296.5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
bnj1296.6 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
bnj1296.7 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
bnj1296.18 (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
bnj1296.9 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1296.10 𝐾 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
bnj1296.11 𝑊 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1296.12 𝐿 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊))}
Assertion
Ref Expression
bnj1296 (𝜓 → (𝑔𝑥) = (𝑥))
Distinct variable groups:   𝐵,𝑓,𝑔   𝐵,,𝑓   𝑥,𝐷   𝐺,𝑑,𝑓,𝑔   ,𝐺,𝑑   𝑊,𝑑,𝑓   𝑔,𝑌   ,𝑌   𝑍,𝑑,𝑓   𝑥,𝑑,𝑓,𝑔   𝑥,
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝜓(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐴(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐷(𝑦,𝑓,𝑔,,𝑑)   𝑅(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐸(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐺(𝑥,𝑦)   𝐾(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝐿(𝑥,𝑦,𝑓,𝑔,,𝑑)   𝑊(𝑥,𝑦,𝑔,)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑔,)

Proof of Theorem bnj1296
StepHypRef Expression
1 bnj1296.18 . . . . 5 (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))
21opeq2d 4546 . . . 4 (𝜓 → ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩)
3 bnj1296.9 . . . 4 𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4 bnj1296.11 . . . 4 𝑊 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩
52, 3, 43eqtr4g 2830 . . 3 (𝜓𝑍 = 𝑊)
65fveq2d 6336 . 2 (𝜓 → (𝐺𝑍) = (𝐺𝑊))
7 bnj1296.7 . . . 4 (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))
8 bnj1296.6 . . . . 5 (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))
9 bnj1296.10 . . . . . . . . . . 11 𝐾 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}
109bnj1436 31248 . . . . . . . . . 10 (𝑔𝐾 → ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
11 fndm 6130 . . . . . . . . . . 11 (𝑔 Fn 𝑑 → dom 𝑔 = 𝑑)
1211anim1i 602 . . . . . . . . . 10 ((𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)) → (dom 𝑔 = 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
1310, 12bnj31 31125 . . . . . . . . 9 (𝑔𝐾 → ∃𝑑𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
14 raleq 3287 . . . . . . . . . . 11 (dom 𝑔 = 𝑑 → (∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍) ↔ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
1514pm5.32i 564 . . . . . . . . . 10 ((dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍)) ↔ (dom 𝑔 = 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
1615rexbii 3189 . . . . . . . . 9 (∃𝑑𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍)) ↔ ∃𝑑𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍)))
1713, 16sylibr 224 . . . . . . . 8 (𝑔𝐾 → ∃𝑑𝐵 (dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍)))
18 simpr 471 . . . . . . . 8 ((dom 𝑔 = 𝑑 ∧ ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍)) → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
1917, 18bnj31 31125 . . . . . . 7 (𝑔𝐾 → ∃𝑑𝐵𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
2019bnj1265 31221 . . . . . 6 (𝑔𝐾 → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
21 bnj1296.2 . . . . . . 7 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
22 bnj1296.3 . . . . . . 7 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
2321, 22, 3, 9bnj1234 31419 . . . . . 6 𝐶 = 𝐾
2420, 23eleq2s 2868 . . . . 5 (𝑔𝐶 → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
258, 24bnj770 31171 . . . 4 (𝜑 → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
267, 25bnj835 31167 . . 3 (𝜓 → ∀𝑥 ∈ dom 𝑔(𝑔𝑥) = (𝐺𝑍))
27 bnj1296.4 . . . . 5 𝐷 = (dom 𝑔 ∩ dom )
2827bnj1292 31224 . . . 4 𝐷 ⊆ dom 𝑔
29 bnj1296.5 . . . . 5 𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}
3029, 7bnj1212 31208 . . . 4 (𝜓𝑥𝐷)
3128, 30bnj1213 31207 . . 3 (𝜓𝑥 ∈ dom 𝑔)
3226, 31bnj1294 31226 . 2 (𝜓 → (𝑔𝑥) = (𝐺𝑍))
33 bnj1296.12 . . . . . . . . . . 11 𝐿 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊))}
3433bnj1436 31248 . . . . . . . . . 10 (𝐿 → ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
35 fndm 6130 . . . . . . . . . . 11 ( Fn 𝑑 → dom = 𝑑)
3635anim1i 602 . . . . . . . . . 10 (( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)) → (dom = 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
3734, 36bnj31 31125 . . . . . . . . 9 (𝐿 → ∃𝑑𝐵 (dom = 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
38 raleq 3287 . . . . . . . . . . 11 (dom = 𝑑 → (∀𝑥 ∈ dom (𝑥) = (𝐺𝑊) ↔ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
3938pm5.32i 564 . . . . . . . . . 10 ((dom = 𝑑 ∧ ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊)) ↔ (dom = 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
4039rexbii 3189 . . . . . . . . 9 (∃𝑑𝐵 (dom = 𝑑 ∧ ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊)) ↔ ∃𝑑𝐵 (dom = 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊)))
4137, 40sylibr 224 . . . . . . . 8 (𝐿 → ∃𝑑𝐵 (dom = 𝑑 ∧ ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊)))
42 simpr 471 . . . . . . . 8 ((dom = 𝑑 ∧ ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊)) → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
4341, 42bnj31 31125 . . . . . . 7 (𝐿 → ∃𝑑𝐵𝑥 ∈ dom (𝑥) = (𝐺𝑊))
4443bnj1265 31221 . . . . . 6 (𝐿 → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
4521, 22, 4, 33bnj1234 31419 . . . . . 6 𝐶 = 𝐿
4644, 45eleq2s 2868 . . . . 5 (𝐶 → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
478, 46bnj771 31172 . . . 4 (𝜑 → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
487, 47bnj835 31167 . . 3 (𝜓 → ∀𝑥 ∈ dom (𝑥) = (𝐺𝑊))
4927bnj1293 31225 . . . 4 𝐷 ⊆ dom
5049, 30bnj1213 31207 . . 3 (𝜓𝑥 ∈ dom )
5148, 50bnj1294 31226 . 2 (𝜓 → (𝑥) = (𝐺𝑊))
526, 32, 513eqtr4d 2815 1 (𝜓 → (𝑔𝑥) = (𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  {crab 3065  cin 3722  wss 3723  cop 4322   class class class wbr 4786  dom cdm 5249  cres 5251   Fn wfn 6026  cfv 6031  w-bnj17 31092   predc-bnj14 31094   FrSe w-bnj15 31098
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-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-bnj17 31093
This theorem is referenced by:  bnj1311  31430
  Copyright terms: Public domain W3C validator