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Theorem ordtypecbv 8305
 Description: Lemma for ordtype 8320. (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 4586 . . . . . . . . . 10 (𝑢 = 𝑟 → (𝑢𝑅𝑣𝑟𝑅𝑣))
43notbid 307 . . . . . . . . 9 (𝑢 = 𝑟 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑟𝑅𝑣))
54cbvralv 3147 . . . . . . . 8 (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑣)
6 breq2 4587 . . . . . . . . . 10 (𝑣 = 𝑠 → (𝑟𝑅𝑣𝑟𝑅𝑠))
76notbid 307 . . . . . . . . 9 (𝑣 = 𝑠 → (¬ 𝑟𝑅𝑣 ↔ ¬ 𝑟𝑅𝑠))
87ralbidv 2969 . . . . . . . 8 (𝑣 = 𝑠 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
95, 8syl5bb 271 . . . . . . 7 (𝑣 = 𝑠 → (∀𝑢𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟𝐶 ¬ 𝑟𝑅𝑠))
109cbvriotav 6522 . . . . . 6 (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠)
11 ordtypelem.2 . . . . . . . . 9 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
12 breq1 4586 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑗𝑅𝑤𝑖𝑅𝑤))
1312cbvralv 3147 . . . . . . . . . . 11 (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑤)
14 breq2 4587 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝑖𝑅𝑤𝑖𝑅𝑦))
1514ralbidv 2969 . . . . . . . . . . 11 (𝑤 = 𝑦 → (∀𝑖 ∈ ran 𝑖𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1613, 15syl5bb 271 . . . . . . . . . 10 (𝑤 = 𝑦 → (∀𝑗 ∈ ran 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran 𝑖𝑅𝑦))
1716cbvrabv 3172 . . . . . . . . 9 {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
1811, 17eqtri 2632 . . . . . . . 8 𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦}
19 rneq 5272 . . . . . . . . . 10 ( = 𝑓 → ran = ran 𝑓)
2019raleqdv 3121 . . . . . . . . 9 ( = 𝑓 → (∀𝑖 ∈ ran 𝑖𝑅𝑦 ↔ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦))
2120rabbidv 3164 . . . . . . . 8 ( = 𝑓 → {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑖𝑅𝑦} = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2218, 21syl5eq 2656 . . . . . . 7 ( = 𝑓𝐶 = {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
2322raleqdv 3121 . . . . . . 7 ( = 𝑓 → (∀𝑟𝐶 ¬ 𝑟𝑅𝑠 ↔ ∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2422, 23riotaeqbidv 6514 . . . . . 6 ( = 𝑓 → (𝑠𝐶𝑟𝐶 ¬ 𝑟𝑅𝑠) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2510, 24syl5eq 2656 . . . . 5 ( = 𝑓 → (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣) = (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
2625cbvmptv 4678 . . . 4 ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣)) = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
272, 26eqtri 2632 . . 3 𝐺 = (𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
28 recseq 7357 . . 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 2633 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 1475  ∀wral 2896  {crab 2900  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ℩crio 6510  recscrecs 7354 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-riota 6511  df-wrecs 7294  df-recs 7355 This theorem is referenced by:  oicl  8317  oif  8318  oiiso2  8319  ordtype  8320  oiiniseg  8321  ordtype2  8322
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