Theorem ordtypecbv 9493

Description: Lemma for ordtype 9508. (Contributed by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
ordtypelem.1	⊢ 𝐹 = recs(𝐺)
ordtypelem.2	⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
ordtypelem.3	⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))

Assertion

Ref	Expression
ordtypecbv	⊢ recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹

Distinct variable groups:   𝑓,𝑟,𝑠,𝑢,𝑣,𝐶   ℎ,𝑗,𝑢,𝑣,𝑤,𝑓,𝑖,𝑦,𝑅,𝑟,𝑠   𝐴,ℎ,𝑗,𝑟,𝑠,𝑢,𝑣,𝑤,𝑦

Allowed substitution hints:   𝐴(𝑓, 𝑖)   𝐶(𝑦, 𝑤, ℎ, 𝑖, 𝑗)   𝐹(𝑦, 𝑤, 𝑣, 𝑢, 𝑓, ℎ, 𝑖, 𝑗, 𝑠, 𝑟)   𝐺(𝑦, 𝑤, 𝑣, 𝑢, 𝑓, ℎ, 𝑖, 𝑗, 𝑠, 𝑟)

Proof of Theorem ordtypecbv

Step	Hyp	Ref	Expression
1		ordtypelem.1	. 2 ⊢ 𝐹 = recs(𝐺)
2		ordtypelem.3	. . . 4 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
3		breq1 5110	. . . . . . . . . 10 ⊢ (𝑢 = 𝑟 → (𝑢𝑅𝑣 ↔ 𝑟𝑅𝑣))
4	3	notbid 321	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑟𝑅𝑣))
5	4	cbvralvw 3242	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑣)
6		breq2 5111	. . . . . . . . . 10 ⊢ (𝑣 = 𝑠 → (𝑟𝑅𝑣 ↔ 𝑟𝑅𝑠))
7	6	notbid 321	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → (¬ 𝑟𝑅𝑣 ↔ ¬ 𝑟𝑅𝑠))
8	7	ralbidv 3187	. . . . . . . 8 ⊢ (𝑣 = 𝑠 → (∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑣 ↔ ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠))
9	5, 8	bitrid 286	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠))
10	9	cbvriotavw 7384	. . . . . 6 ⊢ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣) = (℩𝑠 ∈ 𝐶 ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠)
11		ordtypelem.2	. . . . . . . . 9 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
12		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝑗𝑅𝑤 ↔ 𝑖𝑅𝑤))
13	12	cbvralvw 3242	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑤)
14		breq2 5111	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑖𝑅𝑤 ↔ 𝑖𝑅𝑦))
15	14	ralbidv 3187	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (∀𝑖 ∈ ran ℎ 𝑖𝑅𝑤 ↔ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦))
16	13, 15	bitrid 286	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦))
17	16	cbvrabv 3424	. . . . . . . . 9 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦}
18	11, 17	eqtri 2785	. . . . . . . 8 ⊢ 𝐶 = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦}
19		rneq 5924	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → ran ℎ = ran 𝑓)
20	19	raleqdv 3321	. . . . . . . . 9 ⊢ (ℎ = 𝑓 → (∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦 ↔ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦))
21	20	rabbidv 3421	. . . . . . . 8 ⊢ (ℎ = 𝑓 → {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦} = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
22	18, 21	eqtrid 2809	. . . . . . 7 ⊢ (ℎ = 𝑓 → 𝐶 = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
23	22	raleqdv 3321	. . . . . . 7 ⊢ (ℎ = 𝑓 → (∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠 ↔ ∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
24	22, 23	riotaeqbidv 7377	. . . . . 6 ⊢ (ℎ = 𝑓 → (℩𝑠 ∈ 𝐶 ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠) = (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
25	10, 24	eqtrid 2809	. . . . 5 ⊢ (ℎ = 𝑓 → (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣) = (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
26	25	cbvmptv 5213	. . . 4 ⊢ (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) = (𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
27	2, 26	eqtri 2785	. . 3 ⊢ 𝐺 = (𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
28		recseq 8366	. . 3 ⊢ (𝐺 = (𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠)) → recs(𝐺) = recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))))
29	27, 28	ax-mp 5	. 2 ⊢ recs(𝐺) = recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠)))
30	1, 29	eqtr2i 2786	1 ⊢ recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ∀wral 3078  {crab 3414  Vcvv 3453   class class class wbr 5107   ↦ cmpt 5190  ran crn 5660  ℩crio 7373  recscrecs 8363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-xp 5665  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-iota 6493  df-fv 6545  df-riota 7374  df-ov 7420  df-frecs 8284  df-wrecs 8315  df-recs 8364

This theorem is used by:  oicl  9505  oif  9506  oiiso2  9507  ordtype  9508  oiiniseg  9509  ordtype2  9510