Theorem cantnffval2 9689

Description: An alternate definition of df-cnf 9656 which relies on cantnf 9687. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9658 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}

Assertion

Ref	Expression
cantnffval2	⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐵   𝑤,𝐴,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem cantnffval2

Step	Hyp	Ref	Expression
1		cantnfs.s	. . . . 5 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
2		cantnfs.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ On)
3		cantnfs.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ On)
4		oemapval.t	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
5	1, 2, 3, 4	cantnf 9687	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)))
6		isof1o 7329	. . . 4 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵))
7		f1orel 6825	. . . 4 ⊢ ((𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵) → Rel (𝐴 CNF 𝐵))
8	5, 6, 7	3syl 19	. . 3 ⊢ (𝜑 → Rel (𝐴 CNF 𝐵))
9		dfrel2 6181	. . 3 ⊢ (Rel (𝐴 CNF 𝐵) ↔ ◡◡(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
10	8, 9	sylib 221	. 2 ⊢ (𝜑 → ◡◡(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
11		oecl 8538	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
12	2, 3, 11	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On)
13		eloni 6371	. . . . . 6 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵))
14	12, 13	syl 18	. . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑o 𝐵))
15		isocnv 7336	. . . . . 6 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆))
16	5, 15	syl 18	. . . . 5 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆))
17	1, 2, 3, 4	oemapwe 9688	. . . . . . 7 ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵)))
18	17	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝑇 We 𝑆)
19		ovex 7451	. . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V
20	19	dmex 7919	. . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V
21	1, 20	eqeltri 2857	. . . . . . 7 ⊢ 𝑆 ∈ V
22		exse 5611	. . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆)
23	21, 22	ax-mp 5	. . . . . 6 ⊢ 𝑇 Se 𝑆
24		eqid 2761	. . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆)
25	24	oieu 9526	. . . . . 6 ⊢ ((𝑇 We 𝑆 ∧ 𝑇 Se 𝑆) → ((Ord (𝐴 ↑o 𝐵) ∧ ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) ↔ ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))))
26	18, 23, 25	sylancl 598	. . . . 5 ⊢ (𝜑 → ((Ord (𝐴 ↑o 𝐵) ∧ ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) ↔ ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))))
27	14, 16, 26	mpbi2and 725	. . . 4 ⊢ (𝜑 → ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))
28	27	simprd 501	. . 3 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
29	28	cnveqd 5853	. 2 ⊢ (𝜑 → ◡◡(𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆))
30	10, 29	eqtr3d 2798	1 ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451  {copab 5167   E cep 5550   Se wse 5602   We wwe 5603  ^◡ccnv 5650  dom cdm 5651  Rel wrel 5656  Ord word 6360  Oncon0 6361  –1-1-onto→wf1o 6536  ‘cfv 6537   Isom wiso 6538  (class class class)co 7418   ↑_o coe 8468  OrdIsocoi 9496   CNF ccnf 9655

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-seqom 8451  df-1o 8469  df-2o 8470  df-oadd 8473  df-omul 8474  df-oexp 8475  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-oi 9497  df-cnf 9656

This theorem is used by: (None)