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Theorem hsmexlem2 10498

Description: Lemma for hsmex 10503. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 10653 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 18-Sep-2021.)

**Hypotheses**
Ref	Expression
hsmexlem.f	⊢ 𝐹 = OrdIso( E , 𝐵)
hsmexlem.g	⊢ 𝐺 = OrdIso( E , ∪ 𝑎 ∈ 𝐴 𝐵)

**Assertion**
Ref	Expression
hsmexlem2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))

Distinct variable groups: 𝐴,𝑎 𝐶,𝑎 Allowed substitution hints: 𝐵(𝑎) 𝐹(𝑎) 𝐺(𝑎) 𝑉(𝑎)

Proof of Theorem hsmexlem2 Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4564	. . . . . 6 ⊢ (𝐵 ∈ 𝒫 On → 𝐵 ⊆ On)
2	1	adantr 486	. . . . 5 ⊢ ((𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → 𝐵 ⊆ On)
3	2	ralimi 3100	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → ∀𝑎 ∈ 𝐴 𝐵 ⊆ On)
4		iunss 5003	. . . 4 ⊢ (∪ 𝑎 ∈ 𝐴 𝐵 ⊆ On ↔ ∀𝑎 ∈ 𝐴 𝐵 ⊆ On)
5	3, 4	sylibr 237	. . 3 ⊢ (∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → ∪ 𝑎 ∈ 𝐴 𝐵 ⊆ On)
6	5	3ad2ant3 1153	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → ∪ 𝑎 ∈ 𝐴 𝐵 ⊆ On)
7		xpexg 7762	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On) → (𝐴 × 𝐶) ∈ V)
8	7	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (𝐴 × 𝐶) ∈ V)
9		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑎 𝐶 ∈ On
10		nfra1 3287	. . . . . . . . 9 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)
11	9, 10	nfan 1932	. . . . . . . 8 ⊢ Ⅎ𝑎(𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶))
12		rsp 3251	. . . . . . . . 9 ⊢ (∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → (𝑎 ∈ 𝐴 → (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)))
13		onelss 6404	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On → (dom 𝐹 ∈ 𝐶 → dom 𝐹 ⊆ 𝐶))
14	13	imp 412	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ dom 𝐹 ∈ 𝐶) → dom 𝐹 ⊆ 𝐶)
15	14	adantrl 729	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐹 ⊆ 𝐶)
16	15	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) ∧ 𝑏 ∈ 𝐵) → dom 𝐹 ⊆ 𝐶)
17		hsmexlem.f	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐹 = OrdIso( E , 𝐵)
18	17	oismo 9527	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
19	1, 18	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ 𝒫 On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
20	19	ad2antrl 741	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
21	20	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → ran 𝐹 = 𝐵)
22	17	oif 9517	. . . . . . . . . . . . . . 15 ⊢ 𝐹:dom 𝐹⟶𝐵
23	21, 22	jctil 529	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (𝐹:dom 𝐹⟶𝐵 ∧ ran 𝐹 = 𝐵))
24		dffo2 6798	. . . . . . . . . . . . . 14 ⊢ (𝐹:dom 𝐹–onto→𝐵 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ran 𝐹 = 𝐵))
25	23, 24	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → 𝐹:dom 𝐹–onto→𝐵)
26		dffo3 7100	. . . . . . . . . . . . . 14 ⊢ (𝐹:dom 𝐹–onto→𝐵 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒)))
27	26	simprbi 503	. . . . . . . . . . . . 13 ⊢ (𝐹:dom 𝐹–onto→𝐵 → ∀𝑏 ∈ 𝐵 ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒))
28		rsp 3251	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ 𝐵 ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒) → (𝑏 ∈ 𝐵 → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒)))
29	25, 27, 28	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (𝑏 ∈ 𝐵 → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒)))
30	29	3impia 1135	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) ∧ 𝑏 ∈ 𝐵) → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒))
31		ssrexv 4001	. . . . . . . . . . 11 ⊢ (dom 𝐹 ⊆ 𝐶 → (∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒) → ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒)))
32	16, 30, 31	sylc 66	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) ∧ 𝑏 ∈ 𝐵) → ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒))
33	32	3exp 1137	. . . . . . . . 9 ⊢ (𝐶 ∈ On → ((𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → (𝑏 ∈ 𝐵 → ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒))))
34	12, 33	sylan9r 518	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (𝑎 ∈ 𝐴 → (𝑏 ∈ 𝐵 → ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒))))
35	11, 34	reximdai 3265	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (∃𝑎 ∈ 𝐴 𝑏 ∈ 𝐵 → ∃𝑎 ∈ 𝐴 ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒)))
36	35	3adant1 1148	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (∃𝑎 ∈ 𝐴 𝑏 ∈ 𝐵 → ∃𝑎 ∈ 𝐴 ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒)))
37		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑑∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒)
38		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑎𝐶
39		nfcv 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑎 E
40		nfcsb1v 3871	. . . . . . . . . . 11 ⊢ Ⅎ𝑎⦋𝑑 / 𝑎⦌𝐵
41	39, 40	nfoi 9501	. . . . . . . . . 10 ⊢ Ⅎ𝑎OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)
42		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑎𝑒
43	41, 42	nffv 6893	. . . . . . . . 9 ⊢ Ⅎ𝑎(OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)
44	43	nfeq2 2940	. . . . . . . 8 ⊢ Ⅎ𝑎 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)
45	38, 44	nfrexw 3311	. . . . . . 7 ⊢ Ⅎ𝑎∃𝑒 ∈ 𝐶 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)
46		csbeq1a 3861	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → 𝐵 = ⦋𝑑 / 𝑎⦌𝐵)
47		oieq2 9500	. . . . . . . . . . . 12 ⊢ (𝐵 = ⦋𝑑 / 𝑎⦌𝐵 → OrdIso( E , 𝐵) = OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵))
48	46, 47	syl 18	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑑 → OrdIso( E , 𝐵) = OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵))
49	17, 48	eqtrid 2808	. . . . . . . . . 10 ⊢ (𝑎 = 𝑑 → 𝐹 = OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵))
50	49	fveq1d 6885	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → (𝐹‘𝑒) = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒))
51	50	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → (𝑏 = (𝐹‘𝑒) ↔ 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)))
52	51	rexbidv 3187	. . . . . . 7 ⊢ (𝑎 = 𝑑 → (∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒) ↔ ∃𝑒 ∈ 𝐶 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)))
53	37, 45, 52	cbvrexw 3306	. . . . . 6 ⊢ (∃𝑎 ∈ 𝐴 ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒) ↔ ∃𝑑 ∈ 𝐴 ∃𝑒 ∈ 𝐶 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒))
54	36, 53	imbitrdi 254	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (∃𝑎 ∈ 𝐴 𝑏 ∈ 𝐵 → ∃𝑑 ∈ 𝐴 ∃𝑒 ∈ 𝐶 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)))
55		eliun 4955	. . . . 5 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ 𝐴 𝐵 ↔ ∃𝑎 ∈ 𝐴 𝑏 ∈ 𝐵)
56		vex 3455	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
57		vex 3455	. . . . . . . . . . 11 ⊢ 𝑒 ∈ V
58	56, 57	op1std 8009	. . . . . . . . . 10 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → (1^st ‘𝑐) = 𝑑)
59	58	csbeq1d 3851	. . . . . . . . 9 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → ⦋(1^st ‘𝑐) / 𝑎⦌𝐵 = ⦋𝑑 / 𝑎⦌𝐵)
60		oieq2 9500	. . . . . . . . 9 ⊢ (⦋(1^st ‘𝑐) / 𝑎⦌𝐵 = ⦋𝑑 / 𝑎⦌𝐵 → OrdIso( E , ⦋(1^st ‘𝑐) / 𝑎⦌𝐵) = OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵))
61	59, 60	syl 18	. . . . . . . 8 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → OrdIso( E , ⦋(1^st ‘𝑐) / 𝑎⦌𝐵) = OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵))
62	56, 57	op2ndd 8010	. . . . . . . 8 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → (2^nd ‘𝑐) = 𝑒)
63	61, 62	fveq12d 6890	. . . . . . 7 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → (OrdIso( E , ⦋(1^st ‘𝑐) / 𝑎⦌𝐵)‘(2^nd ‘𝑐)) = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒))
64	63	eqeq2d 2772	. . . . . 6 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → (𝑏 = (OrdIso( E , ⦋(1^st ‘𝑐) / 𝑎⦌𝐵)‘(2^nd ‘𝑐)) ↔ 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)))
65	64	rexxp 5819	. . . . 5 ⊢ (∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , ⦋(1^st ‘𝑐) / 𝑎⦌𝐵)‘(2^nd ‘𝑐)) ↔ ∃𝑑 ∈ 𝐴 ∃𝑒 ∈ 𝐶 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒))
66	54, 55, 65	3imtr4g 299	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (𝑏 ∈ ∪ 𝑎 ∈ 𝐴 𝐵 → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , ⦋(1^st ‘𝑐) / 𝑎⦌𝐵)‘(2^nd ‘𝑐))))
67	66	imp 412	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) ∧ 𝑏 ∈ ∪ 𝑎 ∈ 𝐴 𝐵) → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , ⦋(1^st ‘𝑐) / 𝑎⦌𝐵)‘(2^nd ‘𝑐)))
68	8, 67	wdomd 9568	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → ∪ 𝑎 ∈ 𝐴 𝐵 ≼^* (𝐴 × 𝐶))
69		hsmexlem.g	. . 3 ⊢ 𝐺 = OrdIso( E , ∪ 𝑎 ∈ 𝐴 𝐵)
70	69	hsmexlem1 10497	. 2 ⊢ ((∪ 𝑎 ∈ 𝐴 𝐵 ⊆ On ∧ ∪ 𝑎 ∈ 𝐴 𝐵 ≼^* (𝐴 × 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
71	6, 68, 70	syl2anc 596	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∃wrex 3087 Vcvv 3451 ⦋csb 3847 ⊆ wss 3899 𝒫 cpw 4557 ⟨cop 4590 ∪ ciun 4951 class class class wbr 5103 E cep 5550 × cxp 5649 dom cdm 5651 ran crn 5652 Oncon0 6361 ⟶wf 6533 –onto→wfo 6535 ‘cfv 6537 1^st c1st 7997 2^nd c2nd 7998 Smo wsmo 8346 OrdIsocoi 9496 harchar 9543 ≼^* cwdom 9551

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 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 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-smo 8347 df-recs 8372 df-en 8967 df-dom 8968 df-sdom 8969 df-oi 9497 df-har 9544 df-wdom 9552

This theorem is used by: hsmexlem3 10499

W3C validator