Theorem acwer1prclem 35892

Description: Lemma for acwer1prc 35893. (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 5112	. . . . . . 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 6929	. . . . . . . . 9 ⊢ (𝑅1‘𝐵) ∈ V
5		acnum 35888	. . . . . . . . 9 ⊢ (CHOICE → ((𝑅1‘𝐵) ∈ V → (𝑅1‘𝐵) ∈ dom card))
6	4, 5	mpi 21	. . . . . . . 8 ⊢ (CHOICE → (𝑅1‘𝐵) ∈ dom card)
7		cardid2 10071	. . . . . . . . 9 ⊢ ((𝑅1‘𝐵) ∈ dom card → (card‘(𝑅1‘𝐵)) ≈ (𝑅1‘𝐵))
8		domentr 9078	. . . . . . . . 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 10253	. . . . . 6 ⊢ (CHOICE ↔ ∀𝑧∃𝑦 𝑦 We 𝑧)
16		weeq2 5646	. . . . . . . 8 ⊢ (𝑧 = (𝑅1‘𝐵) → (𝑦 We 𝑧 ↔ 𝑦 We (𝑅1‘𝐵)))
17	16	exbidv 1956	. . . . . . 7 ⊢ (𝑧 = (𝑅1‘𝐵) → (∃𝑦 𝑦 We 𝑧 ↔ ∃𝑦 𝑦 We (𝑅1‘𝐵)))
18	4, 17	spcv 3563	. . . . . 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 35889	. . . . 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 10085	. . . . . 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 1868	. . . 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 6916	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (𝑅1‘𝑥) = (𝑅1‘𝐵))
29	28	sqxpeqd 5690	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → ((𝑅1‘𝑥) × (𝑅1‘𝑥)) = ((𝑅1‘𝐵) × (𝑅1‘𝐵)))
30	29	sseq2d 3945	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ↔ 𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵))))
31		eqidd 2764	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐵 → 𝑠 = 𝑠)
32	31, 28	weeq12d 5647	. . . . . . . . . 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 3580	. . . . . . . 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 6444	. . . . . . . . 9 ⊢ ∅ ∈ On
36		r1fnon 9810	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅1 Fn On
37	36	fndmi 6673	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅1 = On
38	37	eleq2i 2856	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On)
39		ndmfv 6948	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐵 ∈ dom 𝑅1 → (𝑅1‘𝐵) = ∅)
40	38, 39	sylnbir 334	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐵 ∈ On → (𝑅1‘𝐵) = ∅)
41		r10 9812	. . . . . . . . . . . . . 14 ⊢ (𝑅1‘∅) = ∅
42	40, 41	eqtr4di 2816	. . . . . . . . . . . . 13 ⊢ (¬ 𝐵 ∈ On → (𝑅1‘𝐵) = (𝑅1‘∅))
43	42	sqxpeqd 5690	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ On → ((𝑅1‘𝐵) × (𝑅1‘𝐵)) = ((𝑅1‘∅) × (𝑅1‘∅)))
44	43	sseq2d 3945	. . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ On → (𝑠 ⊆ ((𝑅1‘𝐵) × (𝑅1‘𝐵)) ↔ 𝑠 ⊆ ((𝑅1‘∅) × (𝑅1‘∅))))
45		eqidd 2764	. . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ On → 𝑠 = 𝑠)
46	45, 42	weeq12d 5647	. . . . . . . . . . 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 6916	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (𝑅1‘𝑥) = (𝑅1‘∅))
50	49	sqxpeqd 5690	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → ((𝑅1‘𝑥) × (𝑅1‘𝑥)) = ((𝑅1‘∅) × (𝑅1‘∅)))
51	50	sseq2d 3945	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ↔ 𝑠 ⊆ ((𝑅1‘∅) × (𝑅1‘∅))))
52		eqidd 2764	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → 𝑠 = 𝑠)
53	52, 49	weeq12d 5647	. . . . . . . . . . 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 3580	. . . . . . . . 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 3458	. . . . . . . 8 ⊢ 𝑠 ∈ V
59		sseq1 3938	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ↔ 𝑠 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥))))
60		weeq1 5645	. . . . . . . . . 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 3190	. . . . . . . 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 3639	. . . . . . 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 35888	. . . . . . . 8 ⊢ (CHOICE → (𝑠 ∈ 𝑊 → 𝑠 ∈ dom card))
67		cardf2 10061	. . . . . . . . . 10 ⊢ card:{𝑣 ∣ ∃𝑤 ∈ On 𝑤 ≈ 𝑣}⟶On
68		ffun 6742	. . . . . . . . . 10 ⊢ (card:{𝑣 ∣ ∃𝑤 ∈ On 𝑤 ≈ 𝑣}⟶On → Fun card)
69	67, 68	ax-mp 5	. . . . . . . . 9 ⊢ Fun card
70		funfvima 7267	. . . . . . . . 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 2783	. . . . . . . 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 1952	. . 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 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281   class class class wbr 5108   We wwe 5610   × cxp 5656  dom cdm 5658   “ cima 5661  Oncon0 6388  Fun wfun 6560  ⟶wf 6562  ‘cfv 6566  ωcom 7911   ≈ cen 9008   ≼ cdom 9009  𝑅₁cr1 9803  cardccrd 10053  CHOICEwac 10233

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-oi 9541  df-r1 9805  df-card 10057  df-ac 10234

This theorem is used by:  acwer1prc  35893