Theorem cantnflem4 9674

Description: Lemma for cantnf 9675. Complete the induction step of cantnflem3 9673. (Contributed by Mario Carneiro, 25-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)))}
cantnf.c	⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))
cantnf.s	⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
cantnf.e	⊢ (𝜑 → ∅ ∈ 𝐶)
cantnf.x	⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}
cantnf.p	⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
cantnf.y	⊢ 𝑌 = (1st ‘𝑃)
cantnf.z	⊢ 𝑍 = (2nd ‘𝑃)

Assertion

Ref	Expression
cantnflem4	⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))

Distinct variable groups:   𝑤,𝑐,𝑥,𝑦,𝑧,𝐵   𝑎,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧,𝐶   𝐴,𝑎,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐   𝑆,𝑐,𝑥,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑤,𝑌,𝑥,𝑦,𝑧   𝑋,𝑎,𝑏,𝑑,𝑤,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑤, 𝑎, 𝑏, 𝑐, 𝑑)   𝐵(𝑎, 𝑏, 𝑑)   𝑃(𝑥, 𝑦, 𝑧, 𝑤, 𝑎, 𝑏, 𝑐, 𝑑)   𝑆(𝑤, 𝑎, 𝑏, 𝑑)   𝑇(𝑥, 𝑦, 𝑧, 𝑤, 𝑎, 𝑏, 𝑑)   𝑋(𝑐)   𝑌(𝑎, 𝑏, 𝑐, 𝑑)   𝑍(𝑤, 𝑎, 𝑏, 𝑐, 𝑑)

Proof of Theorem cantnflem4

Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnf.s	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
2		cantnfs.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ On)
3		cantnfs.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
4		cantnfs.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ On)
5		oemapval.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
6		cantnf.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))
7		cantnf.e	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ 𝐶)
8	3, 2, 4, 5, 6, 1, 7	cantnflem2 9672	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
9		eqid 2760	. . . . . . . . . . . . . 14 ⊢ 𝑋 = 𝑋
10		eqid 2760	. . . . . . . . . . . . . 14 ⊢ 𝑌 = 𝑌
11		eqid 2760	. . . . . . . . . . . . . 14 ⊢ 𝑍 = 𝑍
12	9, 10, 11	3pm3.2i 1358	. . . . . . . . . . . . 13 ⊢ (𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍)
13		cantnf.x	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}
14		cantnf.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
15		cantnf.y	. . . . . . . . . . . . . 14 ⊢ 𝑌 = (1st ‘𝑃)
16		cantnf.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (2nd ‘𝑃)
17	13, 14, 15, 16	oeeui 8593	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → (((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) ∧ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶) ↔ (𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍)))
18	12, 17	mpbiri 261	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) ∧ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
19	8, 18	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) ∧ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
20	19	simpld 500	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)))
21	20	simp1d 1160	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ On)
22		oecl 8527	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
23	2, 21, 22	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ∈ On)
24	20	simp2d 1161	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 1o))
25	24	eldifad 3911	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
26		onelon 6382	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ On)
27	2, 25, 26	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ On)
28		omcl 8526	. . . . . . . 8 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On)
29	23, 27, 28	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On)
30	20	simp3d 1162	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝐴 ↑o 𝑋))
31		onelon 6382	. . . . . . . 8 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) → 𝑍 ∈ On)
32	23, 30, 31	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ On)
33		oaword1 8542	. . . . . . 7 ⊢ ((((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴 ↑o 𝑋) ·o 𝑌) ⊆ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
34	29, 32, 33	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o 𝑌) ⊆ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
35		dif1o 8490	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (𝐴 ∖ 1o) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ≠ ∅))
36	35	simprbi 503	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐴 ∖ 1o) → 𝑌 ≠ ∅)
37	24, 36	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ ∅)
38		on0eln0 6415	. . . . . . . . . 10 ⊢ (𝑌 ∈ On → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
39	27, 38	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
40	37, 39	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝑌)
41		omword1 8563	. . . . . . . 8 ⊢ ((((𝐴 ↑o 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴 ↑o 𝑋) ⊆ ((𝐴 ↑o 𝑋) ·o 𝑌))
42	23, 27, 40, 41	syl21anc 851	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ⊆ ((𝐴 ↑o 𝑋) ·o 𝑌))
43	42, 30	sseldd 3932	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ((𝐴 ↑o 𝑋) ·o 𝑌))
44	34, 43	sseldd 3932	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
45	19	simprd 501	. . . . 5 ⊢ (𝜑 → (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶)
46	44, 45	eleqtrd 2862	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
47	1, 46	sseldd 3932	. . 3 ⊢ (𝜑 → 𝑍 ∈ ran (𝐴 CNF 𝐵))
48	3, 2, 4	cantnff 9656	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵))
49		ffn 6703	. . . 4 ⊢ ((𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
50		fvelrnb 6939	. . . 4 ⊢ ((𝐴 CNF 𝐵) Fn 𝑆 → (𝑍 ∈ ran (𝐴 CNF 𝐵) ↔ ∃𝑔 ∈ 𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍))
51	48, 49, 50	3syl 19	. . 3 ⊢ (𝜑 → (𝑍 ∈ ran (𝐴 CNF 𝐵) ↔ ∃𝑔 ∈ 𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍))
52	47, 51	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑔 ∈ 𝑆 ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
53	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐴 ∈ On)
54	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐵 ∈ On)
55	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ (𝐴 ↑o 𝐵))
56	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
57	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ∅ ∈ 𝐶)
58		simprl 783	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝑔 ∈ 𝑆)
59		simprr 785	. . 3 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)
60		eqid 2760	. . 3 ⊢ (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝑔‘𝑡))) = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝑔‘𝑡)))
61	3, 53, 54, 5, 55, 56, 57, 13, 14, 15, 16, 58, 59, 60	cantnflem3 9673	. 2 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝑔) = 𝑍)) → 𝐶 ∈ ran (𝐴 CNF 𝐵))
62	52, 61	rexlimddv 3169	1 ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  ifcif 4482  ⟨cop 4590  ∪ cuni 4867  ∩ cint 4907  {copab 5167   ↦ cmpt 5186  dom cdm 5655  ran crn 5656  Oncon0 6357  ℩cio 6487   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986  1_oc1o 8451  2_oc2o 8452   +_o coa 8455   ·_o comu 8456   ↑_o coe 8457   CNF ccnf 9643

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-seqom 8440  df-1o 8458  df-2o 8459  df-oadd 8462  df-omul 8463  df-oexp 8464  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-oi 9485  df-cnf 9644

This theorem is used by:  cantnf  9675