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Theorem r0weon 10026

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 6876	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (1^st ‘𝑥) = (1^st ‘𝑧))
3		fveq2 6876	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (2^nd ‘𝑥) = (2^nd ‘𝑧))
4	2, 3	uneq12d 4144	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)))
5		eqid 2735	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) = (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
6		fvex 6889	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
7		fvex 6889	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑧) ∈ V
8	6, 7	unex 7738	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)) ∈ V
9	4, 5, 8	fvmpt 6986	. . . . . . . . . 10 ⊢ (𝑧 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑧) = ((1^st ‘𝑧) ∪ (2^nd ‘𝑧)))
10		fveq2 6876	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (1^st ‘𝑥) = (1^st ‘𝑤))
11		fveq2 6876	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (2^nd ‘𝑥) = (2^nd ‘𝑤))
12	10, 11	uneq12d 4144	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)))
13		fvex 6889	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑤) ∈ V
14		fvex 6889	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑤) ∈ V
15	13, 14	unex 7738	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)) ∈ V
16	12, 5, 15	fvmpt 6986	. . . . . . . . . 10 ⊢ (𝑤 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))‘𝑤) = ((1^st ‘𝑤) ∪ (2^nd ‘𝑤)))
17	9, 16	breqan12d 5135	. . . . . . . . 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 5555	. . . . . . . . 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 287	. . . . . . . 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 2749	. . . . . . . . 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 631	. . . . . . . 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 918	. . . . . . 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 574	. . . . . 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 5186	. . . . 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 2761	. . . 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 8020	. . . . . . . 8 ⊢ (𝑥 ∈ (On × On) → (1^st ‘𝑥) ∈ On)
27		xp2nd 8021	. . . . . . . 8 ⊢ (𝑥 ∈ (On × On) → (2^nd ‘𝑥) ∈ On)
28		fvex 6889	. . . . . . . . . 10 ⊢ (1^st ‘𝑥) ∈ V
29	28	elon 6361	. . . . . . . . 9 ⊢ ((1^st ‘𝑥) ∈ On ↔ Ord (1^st ‘𝑥))
30		fvex 6889	. . . . . . . . . 10 ⊢ (2^nd ‘𝑥) ∈ V
31	30	elon 6361	. . . . . . . . 9 ⊢ ((2^nd ‘𝑥) ∈ On ↔ Ord (2^nd ‘𝑥))
32		ordun 6458	. . . . . . . . 9 ⊢ ((Ord (1^st ‘𝑥) ∧ Ord (2^nd ‘𝑥)) → Ord ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
33	29, 31, 32	syl2anb 598	. . . . . . . 8 ⊢ (((1^st ‘𝑥) ∈ On ∧ (2^nd ‘𝑥) ∈ On) → Ord ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
34	26, 27, 33	syl2anc 584	. . . . . . 7 ⊢ (𝑥 ∈ (On × On) → Ord ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
35	28, 30	unex 7738	. . . . . . . 8 ⊢ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ V
36	35	elon 6361	. . . . . . 7 ⊢ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ On ↔ Ord ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
37	34, 36	sylibr 234	. . . . . 6 ⊢ (𝑥 ∈ (On × On) → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ On)
38	5, 37	fmpti 7102	. . . . 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 7769	. . . . 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 10025	. . . . 5 ⊢ 𝐿 We (On × On)
44	43	a1i 11	. . . 4 ⊢ (⊤ → 𝐿 We (On × On))
45		vex 3463	. . . . . . . 8 ⊢ 𝑢 ∈ V
46	45	dmex 7905	. . . . . . 7 ⊢ dom 𝑢 ∈ V
47	45	rnex 7906	. . . . . . 7 ⊢ ran 𝑢 ∈ V
48	46, 47	unex 7738	. . . . . 6 ⊢ (dom 𝑢 ∪ ran 𝑢) ∈ V
49		imadmres 6223	. . . . . . 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 4213	. . . . . . . . . 10 ⊢ (𝑢 ∩ (On × On)) ⊆ (On × On)
51		ssun1 4153	. . . . . . . . . . . . . 14 ⊢ dom 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
52		elinel2 4177	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 ∈ (On × On))
53		1st2nd2 8027	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (On × On) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
54	52, 53	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
55		elinel1 4176	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 ∈ 𝑢)
56	54, 55	eqeltrrd 2835	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑢)
57	28, 30	opeldm 5887	. . . . . . . . . . . . . . 15 ⊢ (⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑢 → (1^st ‘𝑥) ∈ dom 𝑢)
58	56, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (1^st ‘𝑥) ∈ dom 𝑢)
59	51, 58	sselid 3956	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (1^st ‘𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
60		ssun2 4154	. . . . . . . . . . . . . 14 ⊢ ran 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
61	28, 30	opelrn 5923	. . . . . . . . . . . . . . 15 ⊢ (⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∈ 𝑢 → (2^nd ‘𝑥) ∈ ran 𝑢)
62	56, 61	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (2^nd ‘𝑥) ∈ ran 𝑢)
63	60, 62	sselid 3956	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (2^nd ‘𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
64	59, 63	prssd 4798	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → {(1^st ‘𝑥), (2^nd ‘𝑥)} ⊆ (dom 𝑢 ∪ ran 𝑢))
65	52, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (1^st ‘𝑥) ∈ On)
66	52, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → (2^nd ‘𝑥) ∈ On)
67		ordunpr 7820	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ {(1^st ‘𝑥), (2^nd ‘𝑥)})
69	64, 68	sseldd 3959	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢))
70	69	rgen 3053	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ (𝑢 ∩ (On × On))((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)
71		ssrab 4048	. . . . . . . . . 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 711	. . . . . . . . 9 ⊢ (𝑢 ∩ (On × On)) ⊆ {𝑥 ∈ (On × On) ∣ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)}
73		dmres 5999	. . . . . . . . . 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 4192	. . . . . . . . . 10 ⊢ (𝑢 ∩ dom (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))) = (𝑢 ∩ (On × On))
76	73, 75	eqtri 2758	. . . . . . . . 9 ⊢ dom ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) = (𝑢 ∩ (On × On))
77	5	mptpreima 6227	. . . . . . . . 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 4010	. . . . . . . 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 5988	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) ↾ 𝑢) ⊆ (𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)))
81		dmss 5882	. . . . . . . . . 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 7044	. . . . . . . . 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 692	. . . . . . . 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 231	. . . . . . 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 4006	. . . . . 6 ⊢ ((𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) ⊆ (dom 𝑢 ∪ ran 𝑢)
87	48, 86	ssexi 5292	. . . . 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 8131	. . 3 ⊢ (⊤ → 𝑅 We (On × On))
90		epse 5636	. . . . 5 ⊢ E Se On
91	90	a1i 11	. . . 4 ⊢ (⊤ → E Se On)
92		vuniex 7733	. . . . . . . 8 ⊢ ∪ 𝑢 ∈ V
93	92	pwex 5350	. . . . . . 7 ⊢ 𝒫 ∪ 𝑢 ∈ V
94	93, 93	xpex 7747	. . . . . 6 ⊢ (𝒫 ∪ 𝑢 × 𝒫 ∪ 𝑢) ∈ V
95	5	mptpreima 6227	. . . . . . . 8 ⊢ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) = {𝑥 ∈ (On × On) ∣ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢}
96		df-rab 3416	. . . . . . . 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 2758	. . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
99		elssuni 4913	. . . . . . . . . . . . 13 ⊢ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢 → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ⊆ ∪ 𝑢)
100	99	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ⊆ ∪ 𝑢)
101	100	unssad 4168	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → (1^st ‘𝑥) ⊆ ∪ 𝑢)
102	28	elpw 4579	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑥) ∈ 𝒫 ∪ 𝑢 ↔ (1^st ‘𝑥) ⊆ ∪ 𝑢)
103	101, 102	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → (1^st ‘𝑥) ∈ 𝒫 ∪ 𝑢)
104	100	unssbd 4169	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → (2^nd ‘𝑥) ⊆ ∪ 𝑢)
105	30	elpw 4579	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑥) ∈ 𝒫 ∪ 𝑢 ↔ (2^nd ‘𝑥) ⊆ ∪ 𝑢)
106	104, 105	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → (2^nd ‘𝑥) ∈ 𝒫 ∪ 𝑢)
107	103, 106	jca 511	. . . . . . . . 9 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → ((1^st ‘𝑥) ∈ 𝒫 ∪ 𝑢 ∧ (2^nd ‘𝑥) ∈ 𝒫 ∪ 𝑢))
108		elxp6 8022	. . . . . . . . 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 583	. . . . . . . 8 ⊢ ((𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢) → 𝑥 ∈ (𝒫 ∪ 𝑢 × 𝒫 ∪ 𝑢))
110	109	abssi 4045	. . . . . . 7 ⊢ {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ 𝑢)} ⊆ (𝒫 ∪ 𝑢 × 𝒫 ∪ 𝑢)
111	97, 110	eqsstri 4005	. . . . . 6 ⊢ (^◡(𝑥 ∈ (On × On) ↦ ((1^st ‘𝑥) ∪ (2^nd ‘𝑥))) “ 𝑢) ⊆ (𝒫 ∪ 𝑢 × 𝒫 ∪ 𝑢)
112	94, 111	ssexi 5292	. . . . 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 8132	. . 3 ⊢ (⊤ → 𝑅 Se (On × On))
115	89, 114	jca 511	. 2 ⊢ (⊤ → (𝑅 We (On × On) ∧ 𝑅 Se (On × On)))
116	115	mptru 1547	1 ⊢ (𝑅 We (On × On) ∧ 𝑅 Se (On × On))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 {cab 2713 ∀wral 3051 {crab 3415 Vcvv 3459 ∪ cun 3924 ∩ cin 3925 ⊆ wss 3926 𝒫 cpw 4575 {cpr 4603 ⟨cop 4607 ∪ cuni 4883 class class class wbr 5119 {copab 5181 ↦ cmpt 5201 E cep 5552 Se wse 5604 We wwe 5605 × cxp 5652 ^◡ccnv 5653 dom cdm 5654 ran crn 5655 ↾ cres 5656 “ cima 5657 Ord word 6351 Oncon0 6352 Fun wfun 6525 ⟶wf 6527 ‘cfv 6531 1^st c1st 7986 2^nd c2nd 7987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-ord 6355 df-on 6356 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-1st 7988 df-2nd 7989

This theorem is referenced by: infxpenlem 10027

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