Theorem cantnflem1b 9582

Description: Lemma for cantnf 9589. (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 772	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (◡𝑂‘𝑋) ⊆ 𝑢)
2		cantnflem1.o	. . . . . . 7 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
3	2	oicl 9421	. . . . . 6 ⊢ Ord dom 𝑂
4		ovexd 7384	. . . . . . . . . 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 9563	. . . . . . . . . . 11 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
10	9	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → E We (𝐺 supp ∅))
11	2	oiiso 9429	. . . . . . . . . 10 ⊢ (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
12	4, 10, 11	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
13		isof1o 7260	. . . . . . . . 9 ⊢ (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅))
15		f1ocnv 6776	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) → ◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
16		f1of 6764	. . . . . . . 8 ⊢ (◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂 → ◡𝑂:(𝐺 supp ∅)⟶dom 𝑂)
17	14, 15, 16	3syl 18	. . . . . . 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 9581	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
23	17, 22	ffvelcdmd 7019	. . . . . 6 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ dom 𝑂)
24		ordelon 6331	. . . . . 6 ⊢ ((Ord dom 𝑂 ∧ (◡𝑂‘𝑋) ∈ dom 𝑂) → (◡𝑂‘𝑋) ∈ On)
25	3, 23, 24	sylancr 587	. . . . 5 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ On)
26	3	a1i 11	. . . . . . . 8 ⊢ (𝜑 → Ord dom 𝑂)
27		ordelon 6331	. . . . . . . 8 ⊢ ((Ord dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
28	26, 27	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
29		onsucb 7750	. . . . . . 7 ⊢ (𝑢 ∈ On ↔ suc 𝑢 ∈ On)
30	28, 29	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ On)
31	30	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑢 ∈ On)
32		ontri1 6341	. . . . 5 ⊢ (((◡𝑂‘𝑋) ∈ On ∧ 𝑢 ∈ On) → ((◡𝑂‘𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (◡𝑂‘𝑋)))
33	25, 31, 32	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → ((◡𝑂‘𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (◡𝑂‘𝑋)))
34	1, 33	mpbid 232	. . 3 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → ¬ 𝑢 ∈ (◡𝑂‘𝑋))
35	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
36		ordtr 6321	. . . . . . . 8 ⊢ (Ord dom 𝑂 → Tr dom 𝑂)
37	3, 36	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → Tr dom 𝑂)
38		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → suc 𝑢 ∈ dom 𝑂)
39		trsuc 6396	. . . . . . 7 ⊢ ((Tr dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ dom 𝑂)
40	37, 38, 39	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑢 ∈ dom 𝑂)
41	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (◡𝑂‘𝑋) ∈ dom 𝑂)
42		isorel 7263	. . . . . 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 836	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑢 E (◡𝑂‘𝑋) ↔ (𝑂‘𝑢) E (𝑂‘(◡𝑂‘𝑋))))
44		fvex 6835	. . . . . 6 ⊢ (◡𝑂‘𝑋) ∈ V
45	44	epeli 5521	. . . . 5 ⊢ (𝑢 E (◡𝑂‘𝑋) ↔ 𝑢 ∈ (◡𝑂‘𝑋))
46		fvex 6835	. . . . . 6 ⊢ (𝑂‘(◡𝑂‘𝑋)) ∈ V
47	46	epeli 5521	. . . . 5 ⊢ ((𝑂‘𝑢) E (𝑂‘(◡𝑂‘𝑋)) ↔ (𝑂‘𝑢) ∈ (𝑂‘(◡𝑂‘𝑋)))
48	43, 45, 47	3bitr3g 313	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑢 ∈ (◡𝑂‘𝑋) ↔ (𝑂‘𝑢) ∈ (𝑂‘(◡𝑂‘𝑋))))
49		f1ocnvfv2 7214	. . . . . . 7 ⊢ ((𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
50	14, 22, 49	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
51	50	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑂‘(◡𝑂‘𝑋)) = 𝑋)
52	51	eleq2d 2814	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → ((𝑂‘𝑢) ∈ (𝑂‘(◡𝑂‘𝑋)) ↔ (𝑂‘𝑢) ∈ 𝑋))
53	48, 52	bitrd 279	. . 3 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑢 ∈ (◡𝑂‘𝑋) ↔ (𝑂‘𝑢) ∈ 𝑋))
54	34, 53	mtbid 324	. 2 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → ¬ (𝑂‘𝑢) ∈ 𝑋)
55	5, 6, 7, 18, 19, 8, 20, 21	oemapvali 9580	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
56	55	simp1d 1142	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
57		onelon 6332	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
58	7, 56, 57	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑋 ∈ On)
59		suppssdm 8110	. . . . . . 7 ⊢ (𝐺 supp ∅) ⊆ dom 𝐺
60	5, 6, 7	cantnfs 9562	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
61	8, 60	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
62	61	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
63	59, 62	fssdm 6671	. . . . . 6 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
64	63	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝐺 supp ∅) ⊆ 𝐵)
65	2	oif 9422	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶(𝐺 supp ∅)
66	65	ffvelcdmi 7017	. . . . . 6 ⊢ (𝑢 ∈ dom 𝑂 → (𝑂‘𝑢) ∈ (𝐺 supp ∅))
67	40, 66	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑂‘𝑢) ∈ (𝐺 supp ∅))
68	64, 67	sseldd 3936	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑂‘𝑢) ∈ 𝐵)
69		onelon 6332	. . . 4 ⊢ ((𝐵 ∈ On ∧ (𝑂‘𝑢) ∈ 𝐵) → (𝑂‘𝑢) ∈ On)
70	7, 68, 69	syl2an2r 685	. . 3 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑂‘𝑢) ∈ On)
71		ontri1 6341	. . 3 ⊢ ((𝑋 ∈ On ∧ (𝑂‘𝑢) ∈ On) → (𝑋 ⊆ (𝑂‘𝑢) ↔ ¬ (𝑂‘𝑢) ∈ 𝑋))
72	58, 70, 71	syl2an2r 685	. 2 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑋 ⊆ (𝑂‘𝑢) ↔ ¬ (𝑂‘𝑢) ∈ 𝑋))
73	54, 72	mpbird 257	1 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3394  Vcvv 3436   ⊆ wss 3903  ∅c0 4284  ∪ cuni 4858   class class class wbr 5092  {copab 5154  Tr wtr 5199   E cep 5518   We wwe 5571  ^◡ccnv 5618  dom cdm 5619  Ord word 6306  Oncon0 6307  suc csuc 6309  ⟶wf 6478  –1-1-onto→wf1o 6481  ‘cfv 6482   Isom wiso 6483  (class class class)co 7349  ωcom 7799   supp csupp 8093   finSupp cfsupp 9251  OrdIsocoi 9401   CNF ccnf 9557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-seqom 8370  df-1o 8388  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-oi 9402  df-cnf 9558

This theorem is referenced by:  cantnflem1c  9583