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Theorem r0weon 9992

Description: A set-like well-ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)

**Hypotheses**
Ref	Expression
leweon.1	⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}
r0weon.1	⊢ 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∨ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))}

**Assertion**
Ref	Expression
r0weon	⊢ (𝑅 We (On × On) ∧ 𝑅 Se (On × On))

Distinct variable groups: 𝑧,𝑤,𝐿 𝑥,𝑤,𝑦,𝑧 Allowed substitution hints: 𝑅(𝑥,𝑦,𝑧,𝑤) 𝐿(𝑥,𝑦)

Proof of Theorem r0weon Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r0weon.1	. . . . 5 ⊢ 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∨ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))}
2		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (1^st ‘𝑥) = (1^st ‘𝑧))
3		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (2^nd ‘𝑥) = (2^nd ‘𝑧))
4	2, 3	uneq12d 4123	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)))
5		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) = (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
6		fvex 6894	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
7		fvex 6894	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑧) ∈ V
8	6, 7	unex 7742	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ V
9	4, 5, 8	fvmpt 6989	. . . . . . . . . 10 ⊢ (𝑧 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) = ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)))
10		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (1^st ‘𝑥) = (1^st ‘𝑤))
11		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (2^nd ‘𝑥) = (2^nd ‘𝑤))
12	10, 11	uneq12d 4123	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)))
13		fvex 6894	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑤) ∈ V
14		fvex 6894	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑤) ∈ V
15	13, 14	unex 7742	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∈ V
16	12, 5, 15	fvmpt 6989	. . . . . . . . . 10 ⊢ (𝑤 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)))
17	9, 16	breqan12d 5125	. . . . . . . . 9 ⊢ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ↔ ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) E ((1^st ‘𝑤) ∪ (2^nd ‘𝑤))))
18	15	epeli 5563	. . . . . . . . 9 ⊢ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) E ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ↔ ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)))
19	17, 18	bitrdi 290	. . . . . . . 8 ⊢ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ↔ ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤))))
20	9, 16	eqeqan12d 2777	. . . . . . . . 9 ⊢ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ↔ ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤))))
21	20	anbi1d 642	. . . . . . . 8 ⊢ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → ((((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤) ↔ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))
22	19, 21	orbi12d 931	. . . . . . 7 ⊢ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → ((((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤)) ↔ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∨ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧𝐿𝑤))))
23	22	pm5.32i 584	. . . . . 6 ⊢ (((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤))) ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∨ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧𝐿𝑤))))
24	23	opabbii 5178	. . . . 5 ⊢ {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤)))} = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∨ (((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))}
25	1, 24	eqtr4i 2789	. . . 4 ⊢ 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤)))}
26		xp1st 8014	. . . . . . . 8 ⊢ (𝑥 ∈ (On × On) → (1^st ‘𝑥) ∈ On)
27		xp2nd 8015	. . . . . . . 8 ⊢ (𝑥 ∈ (On × On) → (2^nd ‘𝑥) ∈ On)
28		fvex 6894	. . . . . . . . . 10 ⊢ (1^st ‘𝑥) ∈ V
29	28	elon 6369	. . . . . . . . 9 ⊢ ((1^st ‘𝑥) ∈ On ↔ Ord (1^st ‘𝑥))
30		fvex 6894	. . . . . . . . . 10 ⊢ (2^nd ‘𝑥) ∈ V
31	30	elon 6369	. . . . . . . . 9 ⊢ ((2^nd ‘𝑥) ∈ On ↔ Ord (2^nd ‘𝑥))
32		ordun 6467	. . . . . . . . 9 ⊢ ((Ord (1^st ‘𝑥) ∧ Ord (2^nd ‘𝑥)) → Ord ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
33	29, 31, 32	syl2anb 609	. . . . . . . 8 ⊢ (((1^st ‘𝑥) ∈ On ∧ (2^nd ‘𝑥) ∈ On) → Ord ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
34	26, 27, 33	syl2anc 595	. . . . . . 7 ⊢ (𝑥 ∈ (On × On) → Ord ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
35	28, 30	unex 7742	. . . . . . . 8 ⊢ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ V
36	35	elon 6369	. . . . . . 7 ⊢ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ On ↔ Ord ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
37	34, 36	sylibr 237	. . . . . 6 ⊢ (𝑥 ∈ (On × On) → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ On)
38	5, 37	fmpti 7107	. . . . 5 ⊢ (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))):(On × On)⟶On
39	38	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))):(On × On)⟶On)
40		epweon 7770	. . . . 5 ⊢ E We On
41	40	a1i 11	. . . 4 ⊢ (⊤ → E We On)
42		leweon.1	. . . . . 6 ⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}
43	42	leweon 9991	. . . . 5 ⊢ 𝐿 We (On × On)
44	43	a1i 11	. . . 4 ⊢ (⊤ → 𝐿 We (On × On))
45		vex 3459	. . . . . . . 8 ⊢ 𝑢 ∈ V
46	45	dmex 7902	. . . . . . 7 ⊢ dom 𝑢 ∈ V
47	45	rnex 7903	. . . . . . 7 ⊢ ran 𝑢 ∈ V
48	46, 47	unex 7742	. . . . . 6 ⊢ (dom 𝑢 ∪ ran 𝑢) ∈ V
49		imadmres 6235	. . . . . . 7 ⊢ ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢)) = ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢)
50		inss2 4190	. . . . . . . . . 10 ⊢ (𝑢 ∩ (On × On)) ⊆ (On × On)
51		ssun1 4131	. . . . . . . . . . . . . 14 ⊢ dom 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
52		elinel2 4155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 ∈ (On × On))
53		1st2nd2 8021	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (On × On) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
54	52, 53	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
55		elinel1 4154	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 ∈ 𝑢)
56	54, 55	eqeltrrd 2864	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑢)
57	28, 30	opeldm 5897	. . . . . . . . . . . . . . 15 ⊢ (⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑢 → (1^st ‘𝑥) ∈ dom 𝑢)
58	56, 57	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (1^st ‘𝑥) ∈ dom 𝑢)
59	51, 58	sselid 3935	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (1^st ‘𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
60		ssun2 4132	. . . . . . . . . . . . . 14 ⊢ ran 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
61	28, 30	opelrn 5933	. . . . . . . . . . . . . . 15 ⊢ (⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑢 → (2^nd ‘𝑥) ∈ ran 𝑢)
62	56, 61	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (2^nd ‘𝑥) ∈ ran 𝑢)
63	60, 62	sselid 3935	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (2^nd ‘𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
64	59, 63	prssd 4788	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → {(1^st ‘𝑥), (2^nd ‘𝑥)} ⊆ (dom 𝑢 ∪ ran 𝑢))
65	52, 26	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (1^st ‘𝑥) ∈ On)
66	52, 27	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (2^nd ‘𝑥) ∈ On)
67		ordunpr 7818	. . . . . . . . . . . . 13 ⊢ (((1^st ‘𝑥) ∈ On ∧ (2^nd ‘𝑥) ∈ On) → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ {(1^st ‘𝑥), (2^nd ‘𝑥)})
68	65, 66, 67	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ {(1^st ‘𝑥), (2^nd ‘𝑥)})
69	64, 68	sseldd 3938	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢))
70	69	rgen 3081	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ (𝑢 ∩ (On × On))((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)
71		ssrab 4025	. . . . . . . . . 10 ⊢ ((𝑢 ∩ (On × On)) ⊆ {𝑥 ∈ (On × On) ∣ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)} ↔ ((𝑢 ∩ (On × On)) ⊆ (On × On) ∧ ∀𝑥 ∈ (𝑢 ∩ (On × On))((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)))
72	50, 70, 71	mpbir2an 723	. . . . . . . . 9 ⊢ (𝑢 ∩ (On × On)) ⊆ {𝑥 ∈ (On × On) ∣ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)}
73		dmres 6011	. . . . . . . . . 10 ⊢ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) = (𝑢 ∩ dom (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))))
74	38	fdmi 6717	. . . . . . . . . . 11 ⊢ dom (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) = (On × On)
75	74	ineq2i 4170	. . . . . . . . . 10 ⊢ (𝑢 ∩ dom (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))) = (𝑢 ∩ (On × On))
76	73, 75	eqtri 2786	. . . . . . . . 9 ⊢ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) = (𝑢 ∩ (On × On))
77	5	mptpreima 6239	. . . . . . . . 9 ⊢ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ (dom 𝑢 ∪ ran 𝑢)) = {𝑥 ∈ (On × On) ∣ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)}
78	72, 76, 77	3sstr4i 3988	. . . . . . . 8 ⊢ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) ⊆ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ (dom 𝑢 ∪ ran 𝑢))
79		funmpt 6574	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
80		resss 6000	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) ⊆ (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
81		dmss 5892	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) ⊆ (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) → dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) ⊆ dom (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))))
82	80, 81	ax-mp 5	. . . . . . . . 9 ⊢ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) ⊆ dom (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
83		funimass3 7049	. . . . . . . . 9 ⊢ ((Fun (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ∧ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) ⊆ dom (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))) → (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢) ↔ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) ⊆ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ (dom 𝑢 ∪ ran 𝑢))))
84	79, 82, 83	mp2an 704	. . . . . . . 8 ⊢ (((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢) ↔ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) ⊆ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ (dom 𝑢 ∪ ran 𝑢)))
85	78, 84	mpbir 234	. . . . . . 7 ⊢ ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢)
86	49, 85	eqsstrri 3984	. . . . . 6 ⊢ ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) ⊆ (dom 𝑢 ∪ ran 𝑢)
87	48, 86	ssexi 5293	. . . . 5 ⊢ ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) ∈ V
88	87	a1i 11	. . . 4 ⊢ (⊤ → ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) ∈ V)
89	25, 39, 41, 44, 88	fnwe 8124	. . 3 ⊢ (⊤ → 𝑅 We (On × On))
90		epse 5643	. . . . 5 ⊢ E Se On
91	90	a1i 11	. . . 4 ⊢ (⊤ → E Se On)
92		vuniex 7737	. . . . . . . 8 ⊢ ∪ 𝑢 ∈ V
93	92	pwex 5351	. . . . . . 7 ⊢ 𝒫 ∪ 𝑢 ∈ V
94	93, 93	xpex 7748	. . . . . 6 ⊢ (𝒫 ∪ 𝑢 × 𝒫 ∪ 𝑢) ∈ V
95	5	mptpreima 6239	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) = {𝑥 ∈ (On × On) ∣ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢}
96		df-rab 3417	. . . . . . . 8 ⊢ {𝑥 ∈ (On × On) ∣ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢} = {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢)}
97	95, 96	eqtri 2786	. . . . . . 7 ⊢ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) = {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢)}
98	53	adantr 485	. . . . . . . . 9 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
99		elssuni 4904	. . . . . . . . . . . . 13 ⊢ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢 → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ⊆ ∪ 𝑢)
100	99	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ⊆ ∪ 𝑢)
101	100	unssad 4146	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → (1^st ‘𝑥) ⊆ ∪ 𝑢)
102	28	elpw 4566	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑥) ∈ 𝒫 ∪ 𝑢 ↔ (1^st ‘𝑥) ⊆ ∪ 𝑢)
103	101, 102	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → (1^st ‘𝑥) ∈ 𝒫 ∪ 𝑢)
104	100	unssbd 4147	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → (2^nd ‘𝑥) ⊆ ∪ 𝑢)
105	30	elpw 4566	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑥) ∈ 𝒫 ∪ 𝑢 ↔ (2^nd ‘𝑥) ⊆ ∪ 𝑢)
106	104, 105	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → (2^nd ‘𝑥) ∈ 𝒫 ∪ 𝑢)
107	103, 106	jca 520	. . . . . . . . 9 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → ((1^st ‘𝑥) ∈ 𝒫 ∪ 𝑢 ∧ (2^nd ‘𝑥) ∈ 𝒫 ∪ 𝑢))
108		elxp6 8016	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ∪ 𝑢 × 𝒫 ∪ 𝑢) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ 𝒫 ∪ 𝑢 ∧ (2^nd ‘𝑥) ∈ 𝒫 ∪ 𝑢)))
109	98, 107, 108	sylanbrc 594	. . . . . . . 8 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → 𝑥 ∈ (𝒫 ∪ 𝑢 × 𝒫 ∪ 𝑢))
110	109	abssi 4022	. . . . . . 7 ⊢ {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢)} ⊆ (𝒫 ∪ 𝑢 × 𝒫 ∪ 𝑢)
111	97, 110	eqsstri 3983	. . . . . 6 ⊢ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) ⊆ (𝒫 ∪ 𝑢 × 𝒫 ∪ 𝑢)
112	94, 111	ssexi 5293	. . . . 5 ⊢ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) ∈ V
113	112	a1i 11	. . . 4 ⊢ (⊤ → (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) ∈ V)
114	25, 39, 91, 113	fnse 8125	. . 3 ⊢ (⊤ → 𝑅 Se (On × On))
115	89, 114	jca 520	. 2 ⊢ (⊤ → (𝑅 We (On × On) ∧ 𝑅 Se (On × On)))
116	115	mptru 1577	1 ⊢ (𝑅 We (On × On) ∧ 𝑅 Se (On × On))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 {cab 2741 ∀wral 3079 {crab 3416 Vcvv 3455 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 𝒫 cpw 4562 {cpr 4591 ⟨cop 4595 ∪ cuni 4872 class class class wbr 5109 {copab 5173 ↦ cmpt 5192 E cep 5560 Se wse 5612 We wwe 5613 × cxp 5659 ^◡ccnv 5660 dom cdm 5661 ran crn 5662 ↾ cres 5663 “ cima 5664 Ord word 6359 Oncon0 6360 Fun wfun 6530 ⟶wf 6532 ‘cfv 6536 1^st c1st 7980 2^nd c2nd 7981

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-ord 6363 df-on 6364 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-1st 7982 df-2nd 7983

This theorem is referenced by: infxpenlem 9993

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