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Theorem hsmexlem2 10496

Description: Lemma for hsmex 10501. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 10644 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 4629	. . . . . 6 ⊢ (𝐵 ∈ 𝒫 On → 𝐵 ⊆ On)
2	1	adantr 480	. . . . 5 ⊢ ((𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → 𝐵 ⊆ On)
3	2	ralimi 3089	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → ∀𝑎 ∈ 𝐴 𝐵 ⊆ On)
4		iunss 5068	. . . 4 ⊢ (∪ 𝑎 ∈ 𝐴 𝐵 ⊆ On ↔ ∀𝑎 ∈ 𝐴 𝐵 ⊆ On)
5	3, 4	sylibr 234	. . 3 ⊢ (∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → ∪ 𝑎 ∈ 𝐴 𝐵 ⊆ On)
6	5	3ad2ant3 1135	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → ∪ 𝑎 ∈ 𝐴 𝐵 ⊆ On)
7		xpexg 7785	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On) → (𝐴 × 𝐶) ∈ V)
8	7	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (𝐴 × 𝐶) ∈ V)
9		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑎 𝐶 ∈ On
10		nfra1 3290	. . . . . . . . 9 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)
11	9, 10	nfan 1898	. . . . . . . 8 ⊢ Ⅎ𝑎(𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶))
12		rsp 3253	. . . . . . . . 9 ⊢ (∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → (𝑎 ∈ 𝐴 → (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)))
13		onelss 6437	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On → (dom 𝐹 ∈ 𝐶 → dom 𝐹 ⊆ 𝐶))
14	13	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ dom 𝐹 ∈ 𝐶) → dom 𝐹 ⊆ 𝐶)
15	14	adantrl 715	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → dom 𝐹 ⊆ 𝐶)
16	15	3adant3 1132	. . . . . . . . . . 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 9609	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
19	1, 18	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ 𝒫 On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
20	19	ad2antrl 727	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
21	20	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → ran 𝐹 = 𝐵)
22	17	oif 9599	. . . . . . . . . . . . . . 15 ⊢ 𝐹:dom 𝐹⟶𝐵
23	21, 22	jctil 519	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (𝐹:dom 𝐹⟶𝐵 ∧ ran 𝐹 = 𝐵))
24		dffo2 6838	. . . . . . . . . . . . . 14 ⊢ (𝐹:dom 𝐹–onto→𝐵 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ran 𝐹 = 𝐵))
25	23, 24	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → 𝐹:dom 𝐹–onto→𝐵)
26		dffo3 7136	. . . . . . . . . . . . . 14 ⊢ (𝐹:dom 𝐹–onto→𝐵 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒)))
27	26	simprbi 496	. . . . . . . . . . . . 13 ⊢ (𝐹:dom 𝐹–onto→𝐵 → ∀𝑏 ∈ 𝐵 ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒))
28		rsp 3253	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ 𝐵 ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒) → (𝑏 ∈ 𝐵 → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒)))
29	25, 27, 28	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (𝑏 ∈ 𝐵 → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒)))
30	29	3impia 1117	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) ∧ 𝑏 ∈ 𝐵) → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒))
31		ssrexv 4078	. . . . . . . . . . 11 ⊢ (dom 𝐹 ⊆ 𝐶 → (∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹‘𝑒) → ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒)))
32	16, 30, 31	sylc 65	. . . . . . . . . 10 ⊢ ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) ∧ 𝑏 ∈ 𝐵) → ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒))
33	32	3exp 1119	. . . . . . . . 9 ⊢ (𝐶 ∈ On → ((𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶) → (𝑏 ∈ 𝐵 → ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒))))
34	12, 33	sylan9r 508	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ ∀𝑎 ∈ 𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐶)) → (𝑎 ∈ 𝐴 → (𝑏 ∈ 𝐵 → ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒))))
35	11, 34	reximdai 3267	. . . . . . 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 1130	. . . . . 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 1913	. . . . . . 7 ⊢ Ⅎ𝑑∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒)
38		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑎𝐶
39		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑎 E
40		nfcsb1v 3946	. . . . . . . . . . 11 ⊢ Ⅎ𝑎⦋𝑑 / 𝑎⦌𝐵
41	39, 40	nfoi 9583	. . . . . . . . . 10 ⊢ Ⅎ𝑎OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)
42		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑎𝑒
43	41, 42	nffv 6930	. . . . . . . . 9 ⊢ Ⅎ𝑎(OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)
44	43	nfeq2 2926	. . . . . . . 8 ⊢ Ⅎ𝑎 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)
45	38, 44	nfrexw 3319	. . . . . . 7 ⊢ Ⅎ𝑎∃𝑒 ∈ 𝐶 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)
46		csbeq1a 3935	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → 𝐵 = ⦋𝑑 / 𝑎⦌𝐵)
47		oieq2 9582	. . . . . . . . . . . 12 ⊢ (𝐵 = ⦋𝑑 / 𝑎⦌𝐵 → OrdIso( E , 𝐵) = OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵))
48	46, 47	syl 17	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑑 → OrdIso( E , 𝐵) = OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵))
49	17, 48	eqtrid 2792	. . . . . . . . . 10 ⊢ (𝑎 = 𝑑 → 𝐹 = OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵))
50	49	fveq1d 6922	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → (𝐹‘𝑒) = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒))
51	50	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → (𝑏 = (𝐹‘𝑒) ↔ 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)))
52	51	rexbidv 3185	. . . . . . 7 ⊢ (𝑎 = 𝑑 → (∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒) ↔ ∃𝑒 ∈ 𝐶 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒)))
53	37, 45, 52	cbvrexw 3313	. . . . . 6 ⊢ (∃𝑎 ∈ 𝐴 ∃𝑒 ∈ 𝐶 𝑏 = (𝐹‘𝑒) ↔ ∃𝑑 ∈ 𝐴 ∃𝑒 ∈ 𝐶 𝑏 = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒))
54	36, 53	imbitrdi 251	. . . . 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 5019	. . . . 5 ⊢ (𝑏 ∈ ∪ 𝑎 ∈ 𝐴 𝐵 ↔ ∃𝑎 ∈ 𝐴 𝑏 ∈ 𝐵)
56		vex 3492	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
57		vex 3492	. . . . . . . . . . 11 ⊢ 𝑒 ∈ V
58	56, 57	op1std 8040	. . . . . . . . . 10 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → (1^st ‘𝑐) = 𝑑)
59	58	csbeq1d 3925	. . . . . . . . 9 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → ⦋(1^st ‘𝑐) / 𝑎⦌𝐵 = ⦋𝑑 / 𝑎⦌𝐵)
60		oieq2 9582	. . . . . . . . 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 17	. . . . . . . 8 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → OrdIso( E , ⦋(1^st ‘𝑐) / 𝑎⦌𝐵) = OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵))
62	56, 57	op2ndd 8041	. . . . . . . 8 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → (2^nd ‘𝑐) = 𝑒)
63	61, 62	fveq12d 6927	. . . . . . 7 ⊢ (𝑐 = ⟨𝑑, 𝑒⟩ → (OrdIso( E , ⦋(1^st ‘𝑐) / 𝑎⦌𝐵)‘(2^nd ‘𝑐)) = (OrdIso( E , ⦋𝑑 / 𝑎⦌𝐵)‘𝑒))
64	63	eqeq2d 2751	. . . . . 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 5867	. . . . 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 296	. . . 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 406	. . 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 9650	. 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 10495	. 2 ⊢ ((∪ 𝑎 ∈ 𝐴 𝐵 ⊆ On ∧ ∪ 𝑎 ∈ 𝐴 𝐵 ≼^* (𝐴 × 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
71	6, 68, 70	syl2anc 583	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

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ⦋csb 3921 ⊆ wss 3976 𝒫 cpw 4622 ⟨cop 4654 ∪ ciun 5015 class class class wbr 5166 E cep 5598 × cxp 5698 dom cdm 5700 ran crn 5701 Oncon0 6395 ⟶wf 6569 –onto→wfo 6571 ‘cfv 6573 1^st c1st 8028 2^nd c2nd 8029 Smo wsmo 8401 OrdIsocoi 9578 harchar 9625 ≼^* cwdom 9633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-smo 8402 df-recs 8427 df-en 9004 df-dom 9005 df-sdom 9006 df-oi 9579 df-har 9626 df-wdom 9634

This theorem is referenced by: hsmexlem3 10497

W3C validator