Theorem ordtypecbv 9480

Description: Lemma for ordtype 9495. (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 5113	. . . . . . . . . 10 ⊢ (𝑢 = 𝑟 → (𝑢𝑅𝑣 ↔ 𝑟𝑅𝑣))
4	3	notbid 321	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (¬ 𝑢𝑅𝑣 ↔ ¬ 𝑟𝑅𝑣))
5	4	cbvralvw 3243	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣 ↔ ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑣)
6		breq2 5114	. . . . . . . . . 10 ⊢ (𝑣 = 𝑠 → (𝑟𝑅𝑣 ↔ 𝑟𝑅𝑠))
7	6	notbid 321	. . . . . . . . 9 ⊢ (𝑣 = 𝑠 → (¬ 𝑟𝑅𝑣 ↔ ¬ 𝑟𝑅𝑠))
8	7	ralbidv 3188	. . . . . . . 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 7379	. . . . . 6 ⊢ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣) = (℩𝑠 ∈ 𝐶 ∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠)
11		ordtypelem.2	. . . . . . . . 9 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
12		breq1 5113	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑖 → (𝑗𝑅𝑤 ↔ 𝑖𝑅𝑤))
13	12	cbvralvw 3243	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤 ↔ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑤)
14		breq2 5114	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑖𝑅𝑤 ↔ 𝑖𝑅𝑦))
15	14	ralbidv 3188	. . . . . . . . . . 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 3426	. . . . . . . . 9 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦}
18	11, 17	eqtri 2786	. . . . . . . 8 ⊢ 𝐶 = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦}
19		rneq 5928	. . . . . . . . . 10 ⊢ (ℎ = 𝑓 → ran ℎ = ran 𝑓)
20	19	raleqdv 3323	. . . . . . . . 9 ⊢ (ℎ = 𝑓 → (∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦 ↔ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦))
21	20	rabbidv 3423	. . . . . . . 8 ⊢ (ℎ = 𝑓 → {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran ℎ 𝑖𝑅𝑦} = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
22	18, 21	eqtrid 2810	. . . . . . 7 ⊢ (ℎ = 𝑓 → 𝐶 = {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦})
23	22	raleqdv 3323	. . . . . . 7 ⊢ (ℎ = 𝑓 → (∀𝑟 ∈ 𝐶 ¬ 𝑟𝑅𝑠 ↔ ∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
24	22, 23	riotaeqbidv 7372	. . . . . 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 2810	. . . . 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 5216	. . . 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 2786	. . 3 ⊢ 𝐺 = (𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))
28		recseq 8361	. . 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 2787	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 1570  ∀wral 3079  {crab 3416  Vcvv 3455   class class class wbr 5110   ↦ cmpt 5193  ran crn 5664  ℩crio 7368  recscrecs 8358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-iota 6494  df-fv 6546  df-riota 7369  df-ov 7415  df-frecs 8279  df-wrecs 8310  df-recs 8359

This theorem is referenced by:  oicl  9492  oif  9493  oiiso2  9494  ordtype  9495  oiiniseg  9496  ordtype2  9497