Theorem cantnflem1b 9666

Description: Lemma for cantnf 9673. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)

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)))}
oemapval.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
oemapval.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
oemapvali.r	⊢ (𝜑 → 𝐹𝑇𝐺)
oemapvali.x	⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
cantnflem1.o	⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))

Assertion

Ref	Expression
cantnflem1b	⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢))

Distinct variable groups:   𝑢,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝐴,𝑐,𝑢,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑢   𝑢,𝐹,𝑤,𝑥,𝑦,𝑧   𝑆,𝑐,𝑢,𝑥,𝑦,𝑧   𝐺,𝑐,𝑢,𝑤,𝑥,𝑦,𝑧   𝑢,𝑂,𝑤,𝑥,𝑦,𝑧   𝜑,𝑢,𝑥,𝑦,𝑧   𝑢,𝑋,𝑤,𝑥,𝑦,𝑧   𝐹,𝑐   𝜑,𝑐

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

Proof of Theorem cantnflem1b

Step	Hyp	Ref	Expression
1		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (◡𝑂‘𝑋) ⊆ 𝑢)
2		cantnflem1.o	. . . . . . 7 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
3	2	oicl 9502	. . . . . 6 ⊢ Ord dom 𝑂
4		ovexd 7449	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 supp ∅) ∈ V)
5		cantnfs.s	. . . . . . . . . . . 12 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
6		cantnfs.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ On)
7		cantnfs.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ On)
8		oemapval.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
9	5, 6, 7, 2, 8	cantnfcl 9647	. . . . . . . . . . 11 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
10	9	simpld 500	. . . . . . . . . 10 ⊢ (𝜑 → E We (𝐺 supp ∅))
11	2	oiiso 9510	. . . . . . . . . 10 ⊢ (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
12	4, 10, 11	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
13		isof1o 7325	. . . . . . . . 9 ⊢ (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
14	12, 13	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
15		f1ocnv 6831	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) → ◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
16		f1of 6818	. . . . . . . 8 ⊢ (◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂 → ◡𝑂:(𝐺 supp ∅)⟶dom 𝑂)
17	14, 15, 16	3syl 19	. . . . . . 7 ⊢ (𝜑 → ◡𝑂:(𝐺 supp ∅)⟶dom 𝑂)
18		oemapval.t	. . . . . . . 8 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
19		oemapval.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
20		oemapvali.r	. . . . . . . 8 ⊢ (𝜑 → 𝐹𝑇𝐺)
21		oemapvali.x	. . . . . . . 8 ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
22	5, 6, 7, 18, 19, 8, 20, 21	cantnflem1a 9665	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
23	17, 22	ffvelcdmd 7079	. . . . . 6 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ dom 𝑂)
24		ordelon 6381	. . . . . 6 ⊢ ((Ord dom 𝑂 ∧ (◡𝑂‘𝑋) ∈ dom 𝑂) → (◡𝑂‘𝑋) ∈ On)
25	3, 23, 24	sylancr 599	. . . . 5 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ On)
26	3	a1i 11	. . . . . . . 8 ⊢ (𝜑 → Ord dom 𝑂)
27		ordelon 6381	. . . . . . . 8 ⊢ ((Ord dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
28	26, 27	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
29		onsucb 7814	. . . . . . 7 ⊢ (𝑢 ∈ On ↔ suc 𝑢 ∈ On)
30	28, 29	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ On)
31	30	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑢 ∈ On)
32		ontri1 6392	. . . . 5 ⊢ (((◡𝑂‘𝑋) ∈ On ∧ 𝑢 ∈ On) → ((◡𝑂‘𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (◡𝑂‘𝑋)))
33	25, 31, 32	syl2an2r 698	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → ((◡𝑂‘𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (◡𝑂‘𝑋)))
34	1, 33	mpbid 235	. . 3 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → ¬ 𝑢 ∈ (◡𝑂‘𝑋))
35	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
36		ordtr 6371	. . . . . . . 8 ⊢ (Ord dom 𝑂 → Tr dom 𝑂)
37	3, 36	mp1i 14	. . . . . . 7 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → Tr dom 𝑂)
38		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → suc 𝑢 ∈ dom 𝑂)
39		trsuc 6447	. . . . . . 7 ⊢ ((Tr dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ dom 𝑂)
40	37, 38, 39	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑢 ∈ dom 𝑂)
41	23	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (◡𝑂‘𝑋) ∈ dom 𝑂)
42		isorel 7328	. . . . . 6 ⊢ ((𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) ∧ (𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ∈ dom 𝑂)) → (𝑢 E (◡𝑂‘𝑋) ↔ (𝑂‘𝑢) E (𝑂‘(◡𝑂‘𝑋))))
43	35, 40, 41, 42	syl12anc 850	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑢 E (◡𝑂‘𝑋) ↔ (𝑂‘𝑢) E (𝑂‘(◡𝑂‘𝑋))))
44		fvex 6892	. . . . . 6 ⊢ (◡𝑂‘𝑋) ∈ V
45	44	epeli 5557	. . . . 5 ⊢ (𝑢 E (◡𝑂‘𝑋) ↔ 𝑢 ∈ (◡𝑂‘𝑋))
46		fvex 6892	. . . . . 6 ⊢ (𝑂‘(◡𝑂‘𝑋)) ∈ V
47	46	epeli 5557	. . . . 5 ⊢ ((𝑂‘𝑢) E (𝑂‘(◡𝑂‘𝑋)) ↔ (𝑂‘𝑢) ∈ (𝑂‘(◡𝑂‘𝑋)))
48	43, 45, 47	3bitr3g 316	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑢 ∈ (◡𝑂‘𝑋) ↔ (𝑂‘𝑢) ∈ (𝑂‘(◡𝑂‘𝑋))))
49		f1ocnvfv2 7279	. . . . . . 7 ⊢ ((𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
50	14, 22, 49	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
51	50	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
52	51	eleq2d 2846	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → ((𝑂‘𝑢) ∈ (𝑂‘(◡𝑂‘𝑋)) ↔ (𝑂‘𝑢) ∈ 𝑋))
53	48, 52	bitrd 282	. . 3 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑢 ∈ (◡𝑂‘𝑋) ↔ (𝑂‘𝑢) ∈ 𝑋))
54	34, 53	mtbid 327	. 2 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → ¬ (𝑂‘𝑢) ∈ 𝑋)
55	5, 6, 7, 18, 19, 8, 20, 21	oemapvali 9664	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
56	55	simp1d 1160	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
57		onelon 6382	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
58	7, 56, 57	syl2anc 596	. . 3 ⊢ (𝜑 → 𝑋 ∈ On)
59		suppssdm 8176	. . . . . . 7 ⊢ (𝐺 supp ∅) ⊆ dom 𝐺
60	5, 6, 7	cantnfs 9646	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
61	8, 60	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
62	61	simpld 500	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
63	59, 62	fssdm 6723	. . . . . 6 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
64	63	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝐺 supp ∅) ⊆ 𝐵)
65	2	oif 9503	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶(𝐺 supp ∅)
66	65	ffvelcdmi 7077	. . . . . 6 ⊢ (𝑢 ∈ dom 𝑂 → (𝑂‘𝑢) ∈ (𝐺 supp ∅))
67	40, 66	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑂‘𝑢) ∈ (𝐺 supp ∅))
68	64, 67	sseldd 3932	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑂‘𝑢) ∈ 𝐵)
69		onelon 6382	. . . 4 ⊢ ((𝐵 ∈ On ∧ (𝑂‘𝑢) ∈ 𝐵) → (𝑂‘𝑢) ∈ On)
70	7, 68, 69	syl2an2r 698	. . 3 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑂‘𝑢) ∈ On)
71		ontri1 6392	. . 3 ⊢ ((𝑋 ∈ On ∧ (𝑂‘𝑢) ∈ On) → (𝑋 ⊆ (𝑂‘𝑢) ↔ ¬ (𝑂‘𝑢) ∈ 𝑋))
72	58, 70, 71	syl2an2r 698	. 2 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑋 ⊆ (𝑂‘𝑢) ↔ ¬ (𝑂‘𝑢) ∈ 𝑋))
73	54, 72	mpbird 260	1 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867   class class class wbr 5103  {copab 5167  Tr wtr 5212   E cep 5554   We wwe 5607  ^◡ccnv 5654  dom cdm 5655  Ord word 6356  Oncon0 6357  suc csuc 6359  ⟶wf 6529  –1-1-onto→wf1o 6532  ‘cfv 6533   Isom wiso 6534  (class class class)co 7414  ωcom 7863   supp csupp 8159   finSupp cfsupp 9332  OrdIsocoi 9482   CNF ccnf 9641

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-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-2nd 7988  df-supp 8160  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-seqom 8438  df-1o 8456  df-map 8829  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-fsupp 9333  df-oi 9483  df-cnf 9642

This theorem is used by:  cantnflem1c  9667