Theorem acwer1prclem 35827

Description: Lemma for acwer1prc 35828. (Contributed by BTernaryTau, 31-Jul-2026.)

Hypothesis

Ref	Expression
acwer1prclem.1	⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}

Assertion

Ref	Expression
acwer1prclem	⊢ ((CHOICE ∧ ω ≼ 𝐴 ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → ∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝐴))

Distinct variable groups:   𝐴,𝑠   𝐵,𝑠,𝑥   𝑠,𝑟,𝑥

Allowed substitution hints:   𝐴(𝑥, 𝑟)   𝐵(𝑟)   𝑊(𝑥, 𝑠, 𝑟)

Proof of Theorem acwer1prclem

Dummy variables 𝑤 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1154	. . 3 ⊢ ((CHOICE ∧ ω ≼ 𝐴 ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → CHOICE)
2		breq2 5106	. . . . . . 7 ⊢ ((card‘(𝑅1‘𝐵)) = 𝐴 → (ω ≼ (card‘(𝑅1‘𝐵)) ↔ ω ≼ 𝐴))
3	2	biimpar 483	. . . . . 6 ⊢ (((card‘(𝑅1‘𝐵)) = 𝐴 ∧ ω ≼ 𝐴) → ω ≼ (card‘(𝑅1‘𝐵)))
4		fvex 6903	. . . . . . . . 9 ⊢ (𝑅1‘𝐵) ∈ V
5		acnum 35823	. . . . . . . . 9 ⊢ (CHOICE → ((𝑅1‘𝐵) ∈ V → (𝑅1‘𝐵) ∈ dom card))
6	4, 5	mpi 21	. . . . . . . 8 ⊢ (CHOICE → (𝑅1‘𝐵) ∈ dom card)
7		cardid2 10041	. . . . . . . . 9 ⊢ ((𝑅1‘𝐵) ∈ dom card → (card‘(𝑅1‘𝐵)) ≈ (𝑅1‘𝐵))
8		domentr 9048	. . . . . . . . 9 ⊢ ((ω ≼ (card‘(𝑅1‘𝐵)) ∧ (card‘(𝑅1‘𝐵)) ≈ (𝑅1‘𝐵)) → ω ≼ (𝑅1‘𝐵))
9	7, 8	sylan2 605	. . . . . . . 8 ⊢ ((ω ≼ (card‘(𝑅1‘𝐵)) ∧ (𝑅1‘𝐵) ∈ dom card) → ω ≼ (𝑅1‘𝐵))
10	6, 9	sylan2 605	. . . . . . 7 ⊢ ((ω ≼ (card‘(𝑅1‘𝐵)) ∧ CHOICE) → ω ≼ (𝑅1‘𝐵))
11	10	expcom 419	. . . . . 6 ⊢ (CHOICE → (ω ≼ (card‘(𝑅1‘𝐵)) → ω ≼ (𝑅1‘𝐵)))
12	3, 11	syl5 35	. . . . 5 ⊢ (CHOICE → (((card‘(𝑅1‘𝐵)) = 𝐴 ∧ ω ≼ 𝐴) → ω ≼ (𝑅1‘𝐵)))
13	12	ancomsd 471	. . . 4 ⊢ (CHOICE → ((ω ≼ 𝐴 ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → ω ≼ (𝑅1‘𝐵)))
14	13	3impib 1134	. . 3 ⊢ ((CHOICE ∧ ω ≼ 𝐴 ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → ω ≼ (𝑅1‘𝐵))
15		dfac8 10221	. . . . . 6 ⊢ (CHOICE ↔ ∀𝑧∃𝑦 𝑦 We 𝑧)
16		weeq2 5639	. . . . . . . 8 ⊢ (𝑧 = (𝑅1‘𝐵) → (𝑦 We 𝑧 ↔ 𝑦 We (𝑅1‘𝐵)))
17	16	exbidv 1953	. . . . . . 7 ⊢ (𝑧 = (𝑅1‘𝐵) → (∃𝑦 𝑦 We 𝑧 ↔ ∃𝑦 𝑦 We (𝑅1‘𝐵)))
18	4, 17	spcv 3559	. . . . . 6 ⊢ (∀𝑧∃𝑦 𝑦 We 𝑧 → ∃𝑦 𝑦 We (𝑅1‘𝐵))
19	15, 18	sylbi 220	. . . . 5 ⊢ (CHOICE → ∃𝑦 𝑦 We (𝑅1‘𝐵))
20		weexenwe 35824	. . . . 5 ⊢ ((∃𝑦 𝑦 We (𝑅1‘𝐵) ∧ ω ≼ (𝑅1‘𝐵)) → ∃𝑠(𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ 𝑠 ≈ (𝑅1‘𝐵)))
21	19, 20	sylan 592	. . . 4 ⊢ ((CHOICE ∧ ω ≼ (𝑅1‘𝐵)) → ∃𝑠(𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ 𝑠 ≈ (𝑅1‘𝐵)))
22		carden2b 10055	. . . . . 6 ⊢ (𝑠 ≈ (𝑅1‘𝐵) → (card‘𝑠) = (card‘(𝑅1‘𝐵)))
23	22	3anim3i 1172	. . . . 5 ⊢ ((𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ 𝑠 ≈ (𝑅1‘𝐵)) → (𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))))
24	23	eximi 1867	. . . 4 ⊢ (∃𝑠(𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ 𝑠 ≈ (𝑅1‘𝐵)) → ∃𝑠(𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))))
25	21, 24	syl 18	. . 3 ⊢ ((CHOICE ∧ ω ≼ (𝑅1‘𝐵)) → ∃𝑠(𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))))
26	1, 14, 25	syl2anc 596	. 2 ⊢ ((CHOICE ∧ ω ≼ 𝐴 ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → ∃𝑠(𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))))
27		df-3an 1105	. . . . 5 ⊢ ((𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))) ↔ ((𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵)) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))))
28		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑅1‘𝑥) = (𝑅1‘𝐵))
29	28	sqxpeqd 5683	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝑅1‘𝑥) × (𝑅1‘𝑥)) = ((𝑅1‘𝐵) × (𝑅1‘𝐵)))
30	29	sseq2d 3941	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ↔ 𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵))))
31		eqidd 2761	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → 𝑠 = 𝑠)
32	31, 28	weeq12d 5640	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑠 We (𝑅1‘𝑥) ↔ 𝑠 We (𝑅1‘𝐵)))
33	30, 32	anbi12d 644	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → ((𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑠 We (𝑅1‘𝑥)) ↔ (𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵))))
34	33	rspcev 3576	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ (𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵))) → ∃𝑥 ∈ On (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑠 We (𝑅1‘𝑥)))
35		0elon 6421	. . . . . . . . 9 ⊢ ∅ ∈ On
36		r1fnon 9780	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅1 Fn On
37	36	fndmi 6648	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅1 = On
38	37	eleq2i 2852	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On)
39		ndmfv 6922	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐵) = ∅)
40	38, 39	sylnbir 334	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐵 ∈ On → (𝑅1‘𝐵) = ∅)
41		r10 9782	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘∅) = ∅
42	40, 41	eqtr4di 2813	. . . . . . . . . . . . 13 ⊢ (¬ 𝐵 ∈ On → (𝑅1‘𝐵) = (𝑅1‘∅))
43	42	sqxpeqd 5683	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ On → ((𝑅1‘𝐵) × (𝑅1‘𝐵)) = ((𝑅1‘∅) × (𝑅1‘∅)))
44	43	sseq2d 3941	. . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ On → (𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ↔ 𝑠 ⊆ ((𝑅1‘∅) × (𝑅1‘∅))))
45		eqidd 2761	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ On → 𝑠 = 𝑠)
46	45, 42	weeq12d 5640	. . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ On → (𝑠 We (𝑅1‘𝐵) ↔ 𝑠 We (𝑅1‘∅)))
47	44, 46	anbi12d 644	. . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ On → ((𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵)) ↔ (𝑠 ⊆ ((𝑅1‘∅) × (𝑅1‘∅)) ∧ 𝑠 We (𝑅1‘∅))))
48	47	biimpa 482	. . . . . . . . 9 ⊢ ((¬ 𝐵 ∈ On ∧ (𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵))) → (𝑠 ⊆ ((𝑅1‘∅) × (𝑅1‘∅)) ∧ 𝑠 We (𝑅1‘∅)))
49		fveq2 6890	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
50	49	sqxpeqd 5683	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) × (𝑅1‘𝑥)) = ((𝑅1‘∅) × (𝑅1‘∅)))
51	50	sseq2d 3941	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ↔ 𝑠 ⊆ ((𝑅1‘∅) × (𝑅1‘∅))))
52		eqidd 2761	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → 𝑠 = 𝑠)
53	52, 49	weeq12d 5640	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑠 We (𝑅1‘𝑥) ↔ 𝑠 We (𝑅1‘∅)))
54	51, 53	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑠 We (𝑅1‘𝑥)) ↔ (𝑠 ⊆ ((𝑅1‘∅) × (𝑅1‘∅)) ∧ 𝑠 We (𝑅1‘∅))))
55	54	rspcev 3576	. . . . . . . . 9 ⊢ ((∅ ∈ On ∧ (𝑠 ⊆ ((𝑅1‘∅) × (𝑅1‘∅)) ∧ 𝑠 We (𝑅1‘∅))) → ∃𝑥 ∈ On (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑠 We (𝑅1‘𝑥)))
56	35, 48, 55	sylancr 599	. . . . . . . 8 ⊢ ((¬ 𝐵 ∈ On ∧ (𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵))) → ∃𝑥 ∈ On (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑠 We (𝑅1‘𝑥)))
57	34, 56	pm2.61ian 824	. . . . . . 7 ⊢ ((𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵)) → ∃𝑥 ∈ On (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑠 We (𝑅1‘𝑥)))
58		vex 3454	. . . . . . . 8 ⊢ 𝑠 ∈ V
59		sseq1 3934	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ↔ 𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥))))
60		weeq1 5638	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 We (𝑅1‘𝑥) ↔ 𝑠 We (𝑅1‘𝑥)))
61	59, 60	anbi12d 644	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → ((𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥)) ↔ (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑠 We (𝑅1‘𝑥))))
62	61	rexbidv 3186	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑠 We (𝑅1‘𝑥))))
63		acwer1prclem.1	. . . . . . . 8 ⊢ 𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}
64	58, 62, 63	elab2 3635	. . . . . . 7 ⊢ (𝑠 ∈ 𝑊 ↔ ∃𝑥 ∈ On (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑠 We (𝑅1‘𝑥)))
65	57, 64	sylibr 237	. . . . . 6 ⊢ ((𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵)) → 𝑠 ∈ 𝑊)
66		acnum 35823	. . . . . . . 8 ⊢ (CHOICE → (𝑠 ∈ 𝑊 → 𝑠 ∈ dom card))
67		cardf2 10031	. . . . . . . . . 10 ⊢ card:{𝑣 ∣ ∃𝑤 ∈ On 𝑤 ≈ 𝑣}⟶On
68		ffun 6717	. . . . . . . . . 10 ⊢ (card:{𝑣 ∣ ∃𝑤 ∈ On 𝑤 ≈ 𝑣}⟶On → Fun card)
69	67, 68	ax-mp 5	. . . . . . . . 9 ⊢ Fun card
70		funfvima 7240	. . . . . . . . 9 ⊢ ((Fun card ∧ 𝑠 ∈ dom card) → (𝑠 ∈ 𝑊 → (card‘𝑠) ∈ (card “ 𝑊)))
71	69, 70	mpan 703	. . . . . . . 8 ⊢ (𝑠 ∈ dom card → (𝑠 ∈ 𝑊 → (card‘𝑠) ∈ (card “ 𝑊)))
72	66, 71	syli 40	. . . . . . 7 ⊢ (CHOICE → (𝑠 ∈ 𝑊 → (card‘𝑠) ∈ (card “ 𝑊)))
73		eqtr 2780	. . . . . . . 8 ⊢ (((card‘𝑠) = (card‘(𝑅1‘𝐵)) ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → (card‘𝑠) = 𝐴)
74	73	expcom 419	. . . . . . 7 ⊢ ((card‘(𝑅1‘𝐵)) = 𝐴 → ((card‘𝑠) = (card‘(𝑅1‘𝐵)) → (card‘𝑠) = 𝐴))
75	72, 74	im2anan9 632	. . . . . 6 ⊢ ((CHOICE ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → ((𝑠 ∈ 𝑊 ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))) → ((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝐴)))
76	65, 75	sylani 616	. . . . 5 ⊢ ((CHOICE ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → (((𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵)) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))) → ((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝐴)))
77	27, 76	biimtrid 245	. . . 4 ⊢ ((CHOICE ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → ((𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))) → ((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝐴)))
78	77	eximdv 1949	. . 3 ⊢ ((CHOICE ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → (∃𝑠(𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))) → ∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝐴)))
79	78	3adant2 1149	. 2 ⊢ ((CHOICE ∧ ω ≼ 𝐴 ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → (∃𝑠(𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ∧ 𝑠 We (𝑅1‘𝐵) ∧ (card‘𝑠) = (card‘(𝑅1‘𝐵))) → ∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝐴)))
80	26, 79	mpd 16	1 ⊢ ((CHOICE ∧ ω ≼ 𝐴 ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → ∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450   ⊆ wss 3896  ∅c0 4277   class class class wbr 5102   We wwe 5603   × cxp 5649  dom cdm 5651   “ cima 5654  Oncon0 6365  Fun wfun 6537  ⟶wf 6539  ‘cfv 6543  ωcom 7881   ≈ cen 8978   ≼ cdom 8979  𝑅₁cr1 9773  cardccrd 10023  CHOICEwac 10201

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-er 8725  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-oi 9511  df-r1 9775  df-card 10027  df-ac 10202

This theorem is used by:  acwer1prc  35828