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Theorem cnfcom2lem 9459

Description: Lemma for cnfcom2 9460. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)

**Hypotheses**
Ref	Expression
cnfcom.s	⊢ 𝑆 = dom (ω CNF 𝐴)
cnfcom.a	⊢ (𝜑 → 𝐴 ∈ On)
cnfcom.b	⊢ (𝜑 → 𝐵 ∈ (ω ↑_o 𝐴))
cnfcom.f	⊢ 𝐹 = (^◡(ω CNF 𝐴)‘𝐵)
cnfcom.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom.h	⊢ 𝐻 = seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +_o 𝑧)), ∅)
cnfcom.t	⊢ 𝑇 = seq_ω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom.m	⊢ 𝑀 = ((ω ↑_o (𝐺‘𝑘)) ·_o (𝐹‘(𝐺‘𝑘)))
cnfcom.k	⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +_o 𝑥)) ∪ ^◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +_o 𝑥)))
cnfcom.w	⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
cnfcom2.1	⊢ (𝜑 → ∅ ∈ 𝐵)

**Assertion**
Ref	Expression
cnfcom2lem	⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)

Distinct variable groups: 𝑥,𝑘,𝑧,𝐴 𝑥,𝑀 𝑓,𝑘,𝑥,𝑧,𝐹 𝑧,𝑇 𝑥,𝑊 𝑓,𝐺,𝑘,𝑥,𝑧 𝑓,𝐻,𝑥 𝑆,𝑘,𝑧 𝜑,𝑘,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑓) 𝐴(𝑓) 𝐵(𝑥,𝑧,𝑓,𝑘) 𝑆(𝑥,𝑓) 𝑇(𝑥,𝑓,𝑘) 𝐻(𝑧,𝑘) 𝐾(𝑥,𝑧,𝑓,𝑘) 𝑀(𝑧,𝑓,𝑘) 𝑊(𝑧,𝑓,𝑘)

**Proof of Theorem cnfcom2lem**
Step	Hyp	Ref	Expression
1		cnfcom2.1	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝐵)
2		n0i 4267	. . . . . 6 ⊢ (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → ¬ 𝐵 = ∅)
4		cnfcom.f	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (^◡(ω CNF 𝐴)‘𝐵)
5		cnfcom.s	. . . . . . . . . . . . . . . . 17 ⊢ 𝑆 = dom (ω CNF 𝐴)
6		omelon 9404	. . . . . . . . . . . . . . . . . 18 ⊢ ω ∈ On
7	6	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ω ∈ On)
8		cnfcom.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ On)
9	5, 7, 8	cantnff1o 9454	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑_o 𝐴))
10		f1ocnv 6728	. . . . . . . . . . . . . . . 16 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑_o 𝐴) → ^◡(ω CNF 𝐴):(ω ↑_o 𝐴)–1-1-onto→𝑆)
11		f1of 6716	. . . . . . . . . . . . . . . 16 ⊢ (^◡(ω CNF 𝐴):(ω ↑_o 𝐴)–1-1-onto→𝑆 → ^◡(ω CNF 𝐴):(ω ↑_o 𝐴)⟶𝑆)
12	9, 10, 11	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ^◡(ω CNF 𝐴):(ω ↑_o 𝐴)⟶𝑆)
13		cnfcom.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ (ω ↑_o 𝐴))
14	12, 13	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
15	4, 14	eqeltrid 2843	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
16	5, 7, 8	cantnfs 9424	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
17	15, 16	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
18	17	simpld 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶ω)
19	18	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → 𝐹:𝐴⟶ω)
20	19	feqmptd 6837	. . . . . . . . 9 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
21		dif0 4306	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ ∅) = 𝐴
22	21	eleq2i 2830	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ ∅) ↔ 𝑥 ∈ 𝐴)
23		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → dom 𝐺 = ∅)
24		ovexd 7310	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V)
25		cnfcom.g	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
26	5, 7, 8, 25, 15	cantnfcl 9425	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
27	26	simpld 495	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → E We (𝐹 supp ∅))
28	25	oien 9297	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → dom 𝐺 ≈ (𝐹 supp ∅))
29	24, 27, 28	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝐺 ≈ (𝐹 supp ∅))
30	29	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → dom 𝐺 ≈ (𝐹 supp ∅))
31	23, 30	eqbrtrrd 5098	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → ∅ ≈ (𝐹 supp ∅))
32	31	ensymd 8791	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → (𝐹 supp ∅) ≈ ∅)
33		en0 8803	. . . . . . . . . . . . . 14 ⊢ ((𝐹 supp ∅) ≈ ∅ ↔ (𝐹 supp ∅) = ∅)
34	32, 33	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → (𝐹 supp ∅) = ∅)
35		ss0b 4331	. . . . . . . . . . . . 13 ⊢ ((𝐹 supp ∅) ⊆ ∅ ↔ (𝐹 supp ∅) = ∅)
36	34, 35	sylibr 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → (𝐹 supp ∅) ⊆ ∅)
37	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → 𝐴 ∈ On)
38		0ex 5231	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → ∅ ∈ V)
40	19, 36, 37, 39	suppssr 8012	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ dom 𝐺 = ∅) ∧ 𝑥 ∈ (𝐴 ∖ ∅)) → (𝐹‘𝑥) = ∅)
41	22, 40	sylan2br 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ dom 𝐺 = ∅) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = ∅)
42	41	mpteq2dva 5174	. . . . . . . . 9 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ ∅))
43	20, 42	eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ ∅))
44		fconstmpt 5649	. . . . . . . 8 ⊢ (𝐴 × {∅}) = (𝑥 ∈ 𝐴 ↦ ∅)
45	43, 44	eqtr4di 2796	. . . . . . 7 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → 𝐹 = (𝐴 × {∅}))
46	45	fveq2d 6778	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → ((ω CNF 𝐴)‘𝐹) = ((ω CNF 𝐴)‘(𝐴 × {∅})))
47	4	fveq2i 6777	. . . . . . . 8 ⊢ ((ω CNF 𝐴)‘𝐹) = ((ω CNF 𝐴)‘(^◡(ω CNF 𝐴)‘𝐵))
48		f1ocnvfv2 7149	. . . . . . . . 9 ⊢ (((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑_o 𝐴) ∧ 𝐵 ∈ (ω ↑_o 𝐴)) → ((ω CNF 𝐴)‘(^◡(ω CNF 𝐴)‘𝐵)) = 𝐵)
49	9, 13, 48	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((ω CNF 𝐴)‘(^◡(ω CNF 𝐴)‘𝐵)) = 𝐵)
50	47, 49	eqtrid 2790	. . . . . . 7 ⊢ (𝜑 → ((ω CNF 𝐴)‘𝐹) = 𝐵)
51	50	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → ((ω CNF 𝐴)‘𝐹) = 𝐵)
52		peano1 7735	. . . . . . . . 9 ⊢ ∅ ∈ ω
53	52	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ ω)
54	5, 7, 8, 53	cantnf0 9433	. . . . . . 7 ⊢ (𝜑 → ((ω CNF 𝐴)‘(𝐴 × {∅})) = ∅)
55	54	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → ((ω CNF 𝐴)‘(𝐴 × {∅})) = ∅)
56	46, 51, 55	3eqtr3d 2786	. . . . 5 ⊢ ((𝜑 ∧ dom 𝐺 = ∅) → 𝐵 = ∅)
57	3, 56	mtand 813	. . . 4 ⊢ (𝜑 → ¬ dom 𝐺 = ∅)
58		nnlim 7726	. . . . 5 ⊢ (dom 𝐺 ∈ ω → ¬ Lim dom 𝐺)
59	26, 58	simpl2im 504	. . . 4 ⊢ (𝜑 → ¬ Lim dom 𝐺)
60		ioran 981	. . . 4 ⊢ (¬ (dom 𝐺 = ∅ ∨ Lim dom 𝐺) ↔ (¬ dom 𝐺 = ∅ ∧ ¬ Lim dom 𝐺))
61	57, 59, 60	sylanbrc 583	. . 3 ⊢ (𝜑 → ¬ (dom 𝐺 = ∅ ∨ Lim dom 𝐺))
62	25	oicl 9288	. . . 4 ⊢ Ord dom 𝐺
63		unizlim 6383	. . . 4 ⊢ (Ord dom 𝐺 → (dom 𝐺 = ∪ dom 𝐺 ↔ (dom 𝐺 = ∅ ∨ Lim dom 𝐺)))
64	62, 63	ax-mp 5	. . 3 ⊢ (dom 𝐺 = ∪ dom 𝐺 ↔ (dom 𝐺 = ∅ ∨ Lim dom 𝐺))
65	61, 64	sylnibr 329	. 2 ⊢ (𝜑 → ¬ dom 𝐺 = ∪ dom 𝐺)
66		orduniorsuc 7677	. . . 4 ⊢ (Ord dom 𝐺 → (dom 𝐺 = ∪ dom 𝐺 ∨ dom 𝐺 = suc ∪ dom 𝐺))
67	62, 66	mp1i 13	. . 3 ⊢ (𝜑 → (dom 𝐺 = ∪ dom 𝐺 ∨ dom 𝐺 = suc ∪ dom 𝐺))
68	67	ord 861	. 2 ⊢ (𝜑 → (¬ dom 𝐺 = ∪ dom 𝐺 → dom 𝐺 = suc ∪ dom 𝐺))
69	65, 68	mpd 15	1 ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∖ cdif 3884 ∪ cun 3885 ⊆ wss 3887 ∅c0 4256 {csn 4561 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 E cep 5494 We wwe 5543 × cxp 5587 ^◡ccnv 5588 dom cdm 5589 Ord word 6265 Oncon0 6266 Lim wlim 6267 suc csuc 6268 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 ωcom 7712 supp csupp 7977 seq_ωcseqom 8278 +_o coa 8294 ·_o comu 8295 ↑_o coe 8296 ≈ cen 8730 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 ax-inf2 9399

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-int 4880 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-1st 7831 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-2o 8298 df-oadd 8301 df-omul 8302 df-oexp 8303 df-er 8498 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: cnfcom2 9460 cnfcom3lem 9461 cnfcom3 9462

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