Theorem cantnflem1c 9445

Description: Lemma for cantnf 9451. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.)

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
cantnflem1c	⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))

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 cantnflem1c

Step	Hyp	Ref	Expression
1		cantnfs.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
2	1	ad3antrrr 727	. 2 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝐵 ∈ On)
3		simplr 766	. 2 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ 𝐵)
4		oemapval.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
5		cantnfs.s	. . . . . . 7 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
6		cantnfs.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
7	5, 6, 1	cantnfs 9424	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
8	4, 7	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
9	8	simpld 495	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
10	9	ffnd 6601	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
11	10	ad3antrrr 727	. 2 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝐺 Fn 𝐵)
12		oemapval.t	. . . . . . 7 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
13		oemapval.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
14		oemapvali.r	. . . . . . 7 ⊢ (𝜑 → 𝐹𝑇𝐺)
15		oemapvali.x	. . . . . . 7 ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
16		cantnflem1.o	. . . . . . 7 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
17	5, 6, 1, 12, 13, 4, 14, 15, 16	cantnflem1b 9444	. . . . . 6 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢))
18	17	ad2antrr 723	. . . . 5 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑋 ⊆ (𝑂‘𝑢))
19		simprr 770	. . . . 5 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝑂‘𝑢) ∈ 𝑥)
20	5, 6, 1, 12, 13, 4, 14, 15	oemapvali 9442	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
21	20	simp1d 1141	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		onelon 6291	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
23	1, 21, 22	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ On)
24	23	ad3antrrr 727	. . . . . 6 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑋 ∈ On)
25		onss 7634	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
26	1, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ On)
27	26	sselda 3921	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
28	27	ad4ant13 748	. . . . . 6 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ On)
29		ontr2 6313	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑥 ∈ On) → ((𝑋 ⊆ (𝑂‘𝑢) ∧ (𝑂‘𝑢) ∈ 𝑥) → 𝑋 ∈ 𝑥))
30	24, 28, 29	syl2anc 584	. . . . 5 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → ((𝑋 ⊆ (𝑂‘𝑢) ∧ (𝑂‘𝑢) ∈ 𝑥) → 𝑋 ∈ 𝑥))
31	18, 19, 30	mp2and 696	. . . 4 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑋 ∈ 𝑥)
32		eleq2w 2822	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑋 ∈ 𝑤 ↔ 𝑋 ∈ 𝑥))
33		fveq2 6774	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
34		fveq2 6774	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝐺‘𝑤) = (𝐺‘𝑥))
35	33, 34	eqeq12d 2754	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐺‘𝑤) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
36	32, 35	imbi12d 345	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ (𝑋 ∈ 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥))))
37	20	simp3d 1143	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
38	37	ad3antrrr 727	. . . . 5 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
39	36, 38, 3	rspcdva 3562	. . . 4 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝑋 ∈ 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥)))
40	31, 39	mpd 15	. . 3 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝐹‘𝑥) = (𝐺‘𝑥))
41		simprl 768	. . 3 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝐹‘𝑥) ≠ ∅)
42	40, 41	eqnetrrd 3012	. 2 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝐺‘𝑥) ≠ ∅)
43		fvn0elsupp 7996	. 2 ⊢ (((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺‘𝑥) ≠ ∅)) → 𝑥 ∈ (𝐺 supp ∅))
44	2, 3, 11, 42, 43	syl22anc 836	1 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839   class class class wbr 5074  {copab 5136   E cep 5494  ^◡ccnv 5588  dom cdm 5589  Oncon0 6266  suc csuc 6268   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   supp csupp 7977   finSupp cfsupp 9128  OrdIsocoi 9268   CNF ccnf 9419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279  df-1o 8297  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-cnf 9420

This theorem is referenced by:  cantnflem1  9447