Theorem brwdom2 9039

Description: Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)

Assertion

Ref	Expression
brwdom2	⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))

Distinct variable groups:   𝑦,𝑋,𝑧   𝑦,𝑌,𝑧

Allowed substitution hints:   𝑉(𝑦,𝑧)

Proof of Theorem brwdom2

Dummy variables 𝑥 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3514	. 2 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ V)
2		0wdom 9036	. . . . . 6 ⊢ (𝑌 ∈ V → ∅ ≼* 𝑌)
3		breq1 5071	. . . . . 6 ⊢ (𝑋 = ∅ → (𝑋 ≼* 𝑌 ↔ ∅ ≼* 𝑌))
4	2, 3	syl5ibrcom 249	. . . . 5 ⊢ (𝑌 ∈ V → (𝑋 = ∅ → 𝑋 ≼* 𝑌))
5	4	imp 409	. . . 4 ⊢ ((𝑌 ∈ V ∧ 𝑋 = ∅) → 𝑋 ≼* 𝑌)
6		0elpw 5258	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝑌
7		f1o0 6653	. . . . . . . 8 ⊢ ∅:∅–1-1-onto→∅
8		f1ofo 6624	. . . . . . . 8 ⊢ (∅:∅–1-1-onto→∅ → ∅:∅–onto→∅)
9		0ex 5213	. . . . . . . . 9 ⊢ ∅ ∈ V
10		foeq1 6588	. . . . . . . . 9 ⊢ (𝑧 = ∅ → (𝑧:∅–onto→∅ ↔ ∅:∅–onto→∅))
11	9, 10	spcev 3609	. . . . . . . 8 ⊢ (∅:∅–onto→∅ → ∃𝑧 𝑧:∅–onto→∅)
12	7, 8, 11	mp2b 10	. . . . . . 7 ⊢ ∃𝑧 𝑧:∅–onto→∅
13		foeq2 6589	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑧:𝑦–onto→∅ ↔ 𝑧:∅–onto→∅))
14	13	exbidv 1922	. . . . . . . 8 ⊢ (𝑦 = ∅ → (∃𝑧 𝑧:𝑦–onto→∅ ↔ ∃𝑧 𝑧:∅–onto→∅))
15	14	rspcev 3625	. . . . . . 7 ⊢ ((∅ ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:∅–onto→∅) → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→∅)
16	6, 12, 15	mp2an 690	. . . . . 6 ⊢ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→∅
17		foeq3 6590	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑦–onto→∅))
18	17	exbidv 1922	. . . . . . 7 ⊢ (𝑋 = ∅ → (∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑧 𝑧:𝑦–onto→∅))
19	18	rexbidv 3299	. . . . . 6 ⊢ (𝑋 = ∅ → (∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→∅))
20	16, 19	mpbiri 260	. . . . 5 ⊢ (𝑋 = ∅ → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)
21	20	adantl 484	. . . 4 ⊢ ((𝑌 ∈ V ∧ 𝑋 = ∅) → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)
22	5, 21	2thd 267	. . 3 ⊢ ((𝑌 ∈ V ∧ 𝑋 = ∅) → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
23		brwdomn0 9035	. . . . 5 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑥 𝑥:𝑌–onto→𝑋))
24	23	adantl 484	. . . 4 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋 ≼* 𝑌 ↔ ∃𝑥 𝑥:𝑌–onto→𝑋))
25		foeq1 6588	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥:𝑌–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋))
26	25	cbvexvw 2044	. . . . . 6 ⊢ (∃𝑥 𝑥:𝑌–onto→𝑋 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)
27		pwidg 4563	. . . . . . . . 9 ⊢ (𝑌 ∈ V → 𝑌 ∈ 𝒫 𝑌)
28	27	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌–onto→𝑋) → 𝑌 ∈ 𝒫 𝑌)
29		foeq2 6589	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋))
30	29	exbidv 1922	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
31	30	rspcev 3625	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝒫 𝑌 ∧ ∃𝑧 𝑧:𝑌–onto→𝑋) → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)
32	28, 31	sylancom 590	. . . . . . 7 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ ∃𝑧 𝑧:𝑌–onto→𝑋) → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)
33	32	ex 415	. . . . . 6 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑧 𝑧:𝑌–onto→𝑋 → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
34	26, 33	syl5bi 244	. . . . 5 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌–onto→𝑋 → ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
35		n0 4312	. . . . . . . . . . 11 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝑋)
36	35	biimpi 218	. . . . . . . . . 10 ⊢ (𝑋 ≠ ∅ → ∃𝑤 𝑤 ∈ 𝑋)
37	36	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) → ∃𝑤 𝑤 ∈ 𝑋)
38		vex 3499	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
39		difexg 5233	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ V → (𝑌 ∖ 𝑦) ∈ V)
40		snex 5334	. . . . . . . . . . . . . 14 ⊢ {𝑤} ∈ V
41		xpexg 7475	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∖ 𝑦) ∈ V ∧ {𝑤} ∈ V) → ((𝑌 ∖ 𝑦) × {𝑤}) ∈ V)
42	39, 40, 41	sylancl 588	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ V → ((𝑌 ∖ 𝑦) × {𝑤}) ∈ V)
43		unexg 7474	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ V ∧ ((𝑌 ∖ 𝑦) × {𝑤}) ∈ V) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) ∈ V)
44	38, 42, 43	sylancr 589	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ V → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) ∈ V)
45	44	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) ∈ V)
46	45	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) ∈ V)
47		fofn 6594	. . . . . . . . . . . . . . 15 ⊢ (𝑧:𝑦–onto→𝑋 → 𝑧 Fn 𝑦)
48	47	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋) → 𝑧 Fn 𝑦)
49	48	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → 𝑧 Fn 𝑦)
50		vex 3499	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
51		fnconstg 6569	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ V → ((𝑌 ∖ 𝑦) × {𝑤}) Fn (𝑌 ∖ 𝑦))
52	50, 51	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ((𝑌 ∖ 𝑦) × {𝑤}) Fn (𝑌 ∖ 𝑦))
53		disjdif 4423	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∩ (𝑌 ∖ 𝑦)) = ∅
54	53	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑦 ∩ (𝑌 ∖ 𝑦)) = ∅)
55		fnun 6465	. . . . . . . . . . . . 13 ⊢ (((𝑧 Fn 𝑦 ∧ ((𝑌 ∖ 𝑦) × {𝑤}) Fn (𝑌 ∖ 𝑦)) ∧ (𝑦 ∩ (𝑌 ∖ 𝑦)) = ∅) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌 ∖ 𝑦)))
56	49, 52, 54, 55	syl21anc 835	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌 ∖ 𝑦)))
57		elpwi 4550	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝒫 𝑌 → 𝑦 ⊆ 𝑌)
58		undif 4432	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝑌 ↔ (𝑦 ∪ (𝑌 ∖ 𝑦)) = 𝑌)
59	57, 58	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 𝑌 → (𝑦 ∪ (𝑌 ∖ 𝑦)) = 𝑌)
60	59	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) → (𝑦 ∪ (𝑌 ∖ 𝑦)) = 𝑌)
61	60	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑦 ∪ (𝑌 ∖ 𝑦)) = 𝑌)
62	61	fneq2d 6449	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ((𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn (𝑦 ∪ (𝑌 ∖ 𝑦)) ↔ (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn 𝑌))
63	56, 62	mpbid 234	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn 𝑌)
64		rnun 6006	. . . . . . . . . . . 12 ⊢ ran (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) = (ran 𝑧 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤}))
65		forn 6595	. . . . . . . . . . . . . . . 16 ⊢ (𝑧:𝑦–onto→𝑋 → ran 𝑧 = 𝑋)
66	65	ad2antll 727	. . . . . . . . . . . . . . 15 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) → ran 𝑧 = 𝑋)
67	66	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ran 𝑧 = 𝑋)
68	67	uneq1d 4140	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (ran 𝑧 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})) = (𝑋 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})))
69		fconst6g 6570	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ 𝑋 → ((𝑌 ∖ 𝑦) × {𝑤}):(𝑌 ∖ 𝑦)⟶𝑋)
70	69	frnd 6523	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝑋 → ran ((𝑌 ∖ 𝑦) × {𝑤}) ⊆ 𝑋)
71	70	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ran ((𝑌 ∖ 𝑦) × {𝑤}) ⊆ 𝑋)
72		ssequn2 4161	. . . . . . . . . . . . . 14 ⊢ (ran ((𝑌 ∖ 𝑦) × {𝑤}) ⊆ 𝑋 ↔ (𝑋 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋)
73	71, 72	sylib 220	. . . . . . . . . . . . 13 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑋 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋)
74	68, 73	eqtrd 2858	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (ran 𝑧 ∪ ran ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋)
75	64, 74	syl5eq 2870	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ran (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋)
76		df-fo 6363	. . . . . . . . . . 11 ⊢ ((𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})):𝑌–onto→𝑋 ↔ ((𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) Fn 𝑌 ∧ ran (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) = 𝑋))
77	63, 75, 76	sylanbrc 585	. . . . . . . . . 10 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})):𝑌–onto→𝑋)
78		foeq1 6588	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})) → (𝑥:𝑌–onto→𝑋 ↔ (𝑧 ∪ ((𝑌 ∖ 𝑦) × {𝑤})):𝑌–onto→𝑋))
79	46, 77, 78	spcedv 3601	. . . . . . . . 9 ⊢ ((((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) ∧ 𝑤 ∈ 𝑋) → ∃𝑥 𝑥:𝑌–onto→𝑋)
80	37, 79	exlimddv 1936	. . . . . . . 8 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ (𝑦 ∈ 𝒫 𝑌 ∧ 𝑧:𝑦–onto→𝑋)) → ∃𝑥 𝑥:𝑌–onto→𝑋)
81	80	expr 459	. . . . . . 7 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (𝑧:𝑦–onto→𝑋 → ∃𝑥 𝑥:𝑌–onto→𝑋))
82	81	exlimdv 1934	. . . . . 6 ⊢ (((𝑌 ∈ V ∧ 𝑋 ≠ ∅) ∧ 𝑦 ∈ 𝒫 𝑌) → (∃𝑧 𝑧:𝑦–onto→𝑋 → ∃𝑥 𝑥:𝑌–onto→𝑋))
83	82	rexlimdva 3286	. . . . 5 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋 → ∃𝑥 𝑥:𝑌–onto→𝑋))
84	34, 83	impbid 214	. . . 4 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌–onto→𝑋 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
85	24, 84	bitrd 281	. . 3 ⊢ ((𝑌 ∈ V ∧ 𝑋 ≠ ∅) → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
86	22, 85	pm2.61dane 3106	. 2 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))
87	1, 86	syl 17	1 ⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∃wrex 3141  Vcvv 3496   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541  {csn 4569   class class class wbr 5068   × cxp 5555  ran crn 5558   Fn wfn 6352  –onto→wfo 6355  –1-1-onto→wf1o 6356   ≼^* cwdom 9023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-wdom 9025

This theorem is referenced by:  brwdom3  9048