MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtypecbv Structured version   Visualization version   GIF version

Theorem ordtypecbv 9586
Description: Lemma for ordtype 9601. (Contributed by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1 𝐹 = recs(𝐺)
ordtypelem.2 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
ordtypelem.3 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
Assertion
Ref Expression
ordtypecbv recs((𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹
Distinct variable groups:   𝑓,𝑟,𝑠,𝑢,𝑣,𝐶   ,𝑗,𝑢,𝑣,𝑤,𝑓,𝑖,𝑦,𝑅,𝑟,𝑠   𝐴,,𝑗,𝑟,𝑠,𝑢,𝑣,𝑤,𝑦
Allowed substitution hints:   𝐴(𝑓,𝑖)   𝐶(𝑦,𝑤,,𝑖,𝑗)   𝐹(𝑦,𝑤,𝑣,𝑢,𝑓,,𝑖,𝑗,𝑠,𝑟)   𝐺(𝑦,𝑤,𝑣,𝑢,𝑓,,𝑖,𝑗,𝑠,𝑟)

Proof of Theorem ordtypecbv
StepHypRef Expression
1 ordtypelem.1 . 2 𝐹 = recs(𝐺)
2 ordtypelem.3 . . . 4 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
3 breq1 5169 . . . . . . . . . 10 (𝑢 = 𝑟 → (𝑢𝑅𝑣𝑟𝑅𝑣))
43notbid 318 . . . . . . . . 9 (𝑢 = 𝑟 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑟𝑅𝑣))
54cbvralvw 3243 . . . . . . . 8 (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑣)
6 breq2 5170 . . . . . . . . . 10 (𝑣 = 𝑠 → (𝑟𝑅𝑣𝑟𝑅𝑠))
76notbid 318 . . . . . . . . 9 (𝑣 = 𝑠 → (¬ 𝑟𝑅𝑣 ↔ ¬ 𝑟𝑅𝑠))
87ralbidv 3184 . . . . . . . 8 (𝑣 = 𝑠 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
95, 8bitrid 283 . . . . . . 7 (𝑣 = 𝑠 → (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
109cbvriotavw 7414 . . . . . 6 (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠)
11 ordtypelem.2 . . . . . . . . 9 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
12 breq1 5169 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑗𝑅𝑤𝑖𝑅𝑤))
1312cbvralvw 3243 . . . . . . . . . . 11 (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑤)
14 breq2 5170 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝑖𝑅𝑤𝑖𝑅𝑦))
1514ralbidv 3184 . . . . . . . . . . 11 (𝑤 = 𝑦 → (∀𝑖 ∈ ran 𝑖𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1613, 15bitrid 283 . . . . . . . . . 10 (𝑤 = 𝑦 → (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1716cbvrabv 3454 . . . . . . . . 9 {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
1811, 17eqtri 2768 . . . . . . . 8 𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
19 rneq 5961 . . . . . . . . . 10 ( = 𝑓 → ran = ran 𝑓)
2019raleqdv 3334 . . . . . . . . 9 ( = 𝑓 → (∀𝑖 ∈ ran 𝑖𝑅𝑦 ↔ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦))
2120rabbidv 3451 . . . . . . . 8 ( = 𝑓 → {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2218, 21eqtrid 2792 . . . . . . 7 ( = 𝑓𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2322raleqdv 3334 . . . . . . 7 ( = 𝑓 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑠 ↔ ∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2422, 23riotaeqbidv 7407 . . . . . 6 ( = 𝑓 → (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2510, 24eqtrid 2792 . . . . 5 ( = 𝑓 → (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2625cbvmptv 5279 . . . 4 ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣)) = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
272, 26eqtri 2768 . . 3 𝐺 = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
28 recseq 8430 . . 3 (𝐺 = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠)) → recs(𝐺) = recs((𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))))
2927, 28ax-mp 5 . 2 recs(𝐺) = recs((𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠)))
301, 29eqtr2i 2769 1 recs((𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wral 3067  {crab 3443  Vcvv 3488   class class class wbr 5166  cmpt 5249  ran crn 5701  crio 7403  recscrecs 8426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fv 6581  df-riota 7404  df-ov 7451  df-frecs 8322  df-wrecs 8353  df-recs 8427
This theorem is referenced by:  oicl  9598  oif  9599  oiiso2  9600  ordtype  9601  oiiniseg  9602  ordtype2  9603
  Copyright terms: Public domain W3C validator