Theorem ordtypecbv 9567

Description: Lemma for ordtype 9582. (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 5159	. . . . . . . . . 10 ⊢ (𝑢 = 𝑟 → (𝑢𝑅𝑣 ↔ 𝑟𝑅𝑣))
4	3	notbid 317	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑟𝑅𝑣))
5	4	cbvralvw 3229	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑣)
6		breq2 5160	. . . . . . . . . 10 ⊢ (𝑣 = 𝑠 → (𝑟𝑅𝑣 ↔ 𝑟𝑅𝑠))
7	6	notbid 317	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → (¬ 𝑟𝑅𝑣 ↔ ¬ 𝑟𝑅𝑠))
8	7	ralbidv 3171	. . . . . . . 8 ⊢ (𝑣 = 𝑠 → (∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑣 ↔ ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠))
9	5, 8	bitrid 282	. . . . . . 7 ⊢ (𝑣 = 𝑠 → (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠))
10	9	cbvriotavw 7396	. . . . . 6 ⊢ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣) = (℩𝑠 ∈ 𝐶 ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠)
11		ordtypelem.2	. . . . . . . . 9 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
12		breq1 5159	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝑗𝑅𝑤 ↔ 𝑖𝑅𝑤))
13	12	cbvralvw 3229	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑤)
14		breq2 5160	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑖𝑅𝑤 ↔ 𝑖𝑅𝑦))
15	14	ralbidv 3171	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (∀𝑖 ∈ ran ℎ 𝑖𝑅𝑤 ↔ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦))
16	13, 15	bitrid 282	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦))
17	16	cbvrabv 3439	. . . . . . . . 9 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦}
18	11, 17	eqtri 2757	. . . . . . . 8 ⊢ 𝐶 = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦}
19		rneq 5946	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → ran ℎ = ran 𝑓)
20	19	raleqdv 3319	. . . . . . . . 9 ⊢ (ℎ = 𝑓 → (∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦 ↔ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦))
21	20	rabbidv 3436	. . . . . . . 8 ⊢ (ℎ = 𝑓 → {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦} = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
22	18, 21	eqtrid 2781	. . . . . . 7 ⊢ (ℎ = 𝑓 → 𝐶 = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
23	22	raleqdv 3319	. . . . . . 7 ⊢ (ℎ = 𝑓 → (∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠 ↔ ∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
24	22, 23	riotaeqbidv 7389	. . . . . 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 2781	. . . . 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 5269	. . . 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 2757	. . 3 ⊢ 𝐺 = (𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
28		recseq 8412	. . 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 2758	1 ⊢ recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹

Syntax hints:  ¬ wn 3   = wceq 1534  ∀wral 3054  {crab 3428  Vcvv 3472   class class class wbr 5156   ↦ cmpt 5239  ran crn 5687  ℩crio 7385  recscrecs 8408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-nul 4336  df-if 4537  df-sn 4637  df-pr 4639  df-op 4643  df-uni 4919  df-br 5157  df-opab 5219  df-mpt 5240  df-xp 5692  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-pred 6317  df-iota 6510  df-fv 6566  df-riota 7386  df-ov 7433  df-frecs 8304  df-wrecs 8335  df-recs 8409

This theorem is referenced by:  oicl  9579  oif  9580  oiiso2  9581  ordtype  9582  oiiniseg  9583  ordtype2  9584