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Theorem ordtypecbv 8967
 Description: Lemma for ordtype 8982. (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 5033 . . . . . . . . . 10 (𝑢 = 𝑟 → (𝑢𝑅𝑣𝑟𝑅𝑣))
43notbid 321 . . . . . . . . 9 (𝑢 = 𝑟 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑟𝑅𝑣))
54cbvralvw 3396 . . . . . . . 8 (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑣)
6 breq2 5034 . . . . . . . . . 10 (𝑣 = 𝑠 → (𝑟𝑅𝑣𝑟𝑅𝑠))
76notbid 321 . . . . . . . . 9 (𝑣 = 𝑠 → (¬ 𝑟𝑅𝑣 ↔ ¬ 𝑟𝑅𝑠))
87ralbidv 3162 . . . . . . . 8 (𝑣 = 𝑠 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
95, 8syl5bb 286 . . . . . . 7 (𝑣 = 𝑠 → (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
109cbvriotavw 7103 . . . . . 6 (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠)
11 ordtypelem.2 . . . . . . . . 9 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
12 breq1 5033 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑗𝑅𝑤𝑖𝑅𝑤))
1312cbvralvw 3396 . . . . . . . . . . 11 (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑤)
14 breq2 5034 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝑖𝑅𝑤𝑖𝑅𝑦))
1514ralbidv 3162 . . . . . . . . . . 11 (𝑤 = 𝑦 → (∀𝑖 ∈ ran 𝑖𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1613, 15syl5bb 286 . . . . . . . . . 10 (𝑤 = 𝑦 → (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1716cbvrabv 3439 . . . . . . . . 9 {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
1811, 17eqtri 2821 . . . . . . . 8 𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
19 rneq 5770 . . . . . . . . . 10 ( = 𝑓 → ran = ran 𝑓)
2019raleqdv 3364 . . . . . . . . 9 ( = 𝑓 → (∀𝑖 ∈ ran 𝑖𝑅𝑦 ↔ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦))
2120rabbidv 3427 . . . . . . . 8 ( = 𝑓 → {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2218, 21syl5eq 2845 . . . . . . 7 ( = 𝑓𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2322raleqdv 3364 . . . . . . 7 ( = 𝑓 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑠 ↔ ∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2422, 23riotaeqbidv 7096 . . . . . 6 ( = 𝑓 → (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2510, 24syl5eq 2845 . . . . 5 ( = 𝑓 → (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2625cbvmptv 5133 . . . 4 ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣)) = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
272, 26eqtri 2821 . . 3 𝐺 = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
28 recseq 7995 . . 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 2822 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 1538  ∀wral 3106  {crab 3110  Vcvv 3441   class class class wbr 5030   ↦ cmpt 5110  ran crn 5520  ℩crio 7092  recscrecs 7992 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fv 6332  df-riota 7093  df-wrecs 7932  df-recs 7993 This theorem is referenced by:  oicl  8979  oif  8980  oiiso2  8981  ordtype  8982  oiiniseg  8983  ordtype2  8984
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