Theorem cantnflem1b 9576

Description: Lemma for cantnf 9583. (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 9415	. . . . . 6 ⊢ Ord dom 𝑂
4		ovexd 7381	. . . . . . . . . 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 9557	. . . . . . . . . . 11 ⊢ (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
10	9	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → E We (𝐺 supp ∅))
11	2	oiiso 9423	. . . . . . . . . 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 7257	. . . . . . . . 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 6775	. . . . . . . 8 ⊢ (𝑂:dom 𝑂–1-1-onto→(𝐺 supp ∅) → ◡𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
16		f1of 6763	. . . . . . . 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 9575	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
23	17, 22	ffvelcdmd 7018	. . . . . 6 ⊢ (𝜑 → (◡𝑂‘𝑋) ∈ dom 𝑂)
24		ordelon 6330	. . . . . 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 6330	. . . . . . . 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 7747	. . . . . . 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 6340	. . . . 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 6320	. . . . . . . 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 6395	. . . . . . 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 7260	. . . . . 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 5516	. . . . 5 ⊢ (𝑢 E (◡𝑂‘𝑋) ↔ 𝑢 ∈ (◡𝑂‘𝑋))
46		fvex 6835	. . . . . 6 ⊢ (𝑂‘(◡𝑂‘𝑋)) ∈ V
47	46	epeli 5516	. . . . 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 7211	. . . . . . 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 2817	. . . 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 9574	. . . . 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 6331	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
58	7, 56, 57	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑋 ∈ On)
59		suppssdm 8107	. . . . . . 7 ⊢ (𝐺 supp ∅) ⊆ dom 𝐺
60	5, 6, 7	cantnfs 9556	. . . . . . . . 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 6670	. . . . . 6 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
64	63	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝐺 supp ∅) ⊆ 𝐵)
65	2	oif 9416	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶(𝐺 supp ∅)
66	65	ffvelcdmi 7016	. . . . . 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 3930	. . . 4 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → (𝑂‘𝑢) ∈ 𝐵)
69		onelon 6331	. . . 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 6340	. . 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  ∪ cuni 4856   class class class wbr 5089  {copab 5151  Tr wtr 5196   E cep 5513   We wwe 5566  ^◡ccnv 5613  dom cdm 5614  Ord word 6305  Oncon0 6306  suc csuc 6308  ⟶wf 6477  –1-1-onto→wf1o 6480  ‘cfv 6481   Isom wiso 6482  (class class class)co 7346  ωcom 7796   supp csupp 8090   finSupp cfsupp 9245  OrdIsocoi 9395   CNF ccnf 9551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-seqom 8367  df-1o 8385  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-oi 9396  df-cnf 9552

This theorem is referenced by:  cantnflem1c  9577