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Theorem ordtypecbv 8577
Description: Lemma for ordtype 8592. (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 4787 . . . . . . . . . 10 (𝑢 = 𝑟 → (𝑢𝑅𝑣𝑟𝑅𝑣))
43notbid 307 . . . . . . . . 9 (𝑢 = 𝑟 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑟𝑅𝑣))
54cbvralv 3319 . . . . . . . 8 (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑣)
6 breq2 4788 . . . . . . . . . 10 (𝑣 = 𝑠 → (𝑟𝑅𝑣𝑟𝑅𝑠))
76notbid 307 . . . . . . . . 9 (𝑣 = 𝑠 → (¬ 𝑟𝑅𝑣 ↔ ¬ 𝑟𝑅𝑠))
87ralbidv 3134 . . . . . . . 8 (𝑣 = 𝑠 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
95, 8syl5bb 272 . . . . . . 7 (𝑣 = 𝑠 → (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
109cbvriotav 6764 . . . . . 6 (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠)
11 ordtypelem.2 . . . . . . . . 9 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
12 breq1 4787 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑗𝑅𝑤𝑖𝑅𝑤))
1312cbvralv 3319 . . . . . . . . . . 11 (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑤)
14 breq2 4788 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝑖𝑅𝑤𝑖𝑅𝑦))
1514ralbidv 3134 . . . . . . . . . . 11 (𝑤 = 𝑦 → (∀𝑖 ∈ ran 𝑖𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1613, 15syl5bb 272 . . . . . . . . . 10 (𝑤 = 𝑦 → (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1716cbvrabv 3348 . . . . . . . . 9 {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
1811, 17eqtri 2792 . . . . . . . 8 𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
19 rneq 5489 . . . . . . . . . 10 ( = 𝑓 → ran = ran 𝑓)
2019raleqdv 3292 . . . . . . . . 9 ( = 𝑓 → (∀𝑖 ∈ ran 𝑖𝑅𝑦 ↔ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦))
2120rabbidv 3338 . . . . . . . 8 ( = 𝑓 → {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2218, 21syl5eq 2816 . . . . . . 7 ( = 𝑓𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2322raleqdv 3292 . . . . . . 7 ( = 𝑓 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑠 ↔ ∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2422, 23riotaeqbidv 6756 . . . . . 6 ( = 𝑓 → (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2510, 24syl5eq 2816 . . . . 5 ( = 𝑓 → (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2625cbvmptv 4882 . . . 4 ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣)) = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
272, 26eqtri 2792 . . 3 𝐺 = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
28 recseq 7622 . . 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 2793 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 1630  wral 3060  {crab 3064  Vcvv 3349   class class class wbr 4784  cmpt 4861  ran crn 5250  crio 6752  recscrecs 7619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-iota 5994  df-fv 6039  df-riota 6753  df-wrecs 7558  df-recs 7620
This theorem is referenced by:  oicl  8589  oif  8590  oiiso2  8591  ordtype  8592  oiiniseg  8593  ordtype2  8594
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