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Theorem ordtypecbv 9133
Description: Lemma for ordtype 9148. (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 5056 . . . . . . . . . 10 (𝑢 = 𝑟 → (𝑢𝑅𝑣𝑟𝑅𝑣))
43notbid 321 . . . . . . . . 9 (𝑢 = 𝑟 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑟𝑅𝑣))
54cbvralvw 3358 . . . . . . . 8 (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑣)
6 breq2 5057 . . . . . . . . . 10 (𝑣 = 𝑠 → (𝑟𝑅𝑣𝑟𝑅𝑠))
76notbid 321 . . . . . . . . 9 (𝑣 = 𝑠 → (¬ 𝑟𝑅𝑣 ↔ ¬ 𝑟𝑅𝑠))
87ralbidv 3118 . . . . . . . 8 (𝑣 = 𝑠 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
95, 8syl5bb 286 . . . . . . 7 (𝑣 = 𝑠 → (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
109cbvriotavw 7180 . . . . . 6 (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠)
11 ordtypelem.2 . . . . . . . . 9 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
12 breq1 5056 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑗𝑅𝑤𝑖𝑅𝑤))
1312cbvralvw 3358 . . . . . . . . . . 11 (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑤)
14 breq2 5057 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝑖𝑅𝑤𝑖𝑅𝑦))
1514ralbidv 3118 . . . . . . . . . . 11 (𝑤 = 𝑦 → (∀𝑖 ∈ ran 𝑖𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1613, 15syl5bb 286 . . . . . . . . . 10 (𝑤 = 𝑦 → (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1716cbvrabv 3402 . . . . . . . . 9 {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
1811, 17eqtri 2765 . . . . . . . 8 𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
19 rneq 5805 . . . . . . . . . 10 ( = 𝑓 → ran = ran 𝑓)
2019raleqdv 3325 . . . . . . . . 9 ( = 𝑓 → (∀𝑖 ∈ ran 𝑖𝑅𝑦 ↔ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦))
2120rabbidv 3390 . . . . . . . 8 ( = 𝑓 → {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2218, 21eqtrid 2789 . . . . . . 7 ( = 𝑓𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2322raleqdv 3325 . . . . . . 7 ( = 𝑓 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑠 ↔ ∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2422, 23riotaeqbidv 7173 . . . . . 6 ( = 𝑓 → (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2510, 24eqtrid 2789 . . . . 5 ( = 𝑓 → (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2625cbvmptv 5158 . . . 4 ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣)) = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
272, 26eqtri 2765 . . 3 𝐺 = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
28 recseq 8110 . . 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 2766 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 1543  wral 3061  {crab 3065  Vcvv 3408   class class class wbr 5053  cmpt 5135  ran crn 5552  crio 7169  recscrecs 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-xp 5557  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-iota 6338  df-fv 6388  df-riota 7170  df-wrecs 8047  df-recs 8108
This theorem is referenced by:  oicl  9145  oif  9146  oiiso2  9147  ordtype  9148  oiiniseg  9149  ordtype2  9150
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